Size effects on stress concentration induced by a prolate ellipsoidal particle and void nucleation mechanism

Abstract

There generally exist two void nucleation mechanisms in materials, i.e. the breakage of hard second-phase particle and the separation of particle–matrix interface. The role of particle shape in governing the void nucleation mechanism has already been investigated carefully in the literatures. In this study, the coupled effects of particle size and shape on the void nucleation mechanisms, which have not yet been carefully addressed, have been paid to special attention. To this end, a wide range of particle aspect ratios (but limited to the prolate spheroidal particle) is considered to reflect the shape effect; and the size effect is captured by the Fleck–Hutchinson phenomenological strain plasticity constitutive theory (Advance in Applied Mechanics, vol. 33, Academic Press, New York, 1997, p. 295). Detailed theoretical analyses and computations on an infinite block containing an isolated elastic prolate spheroidal particle are carried out to light the features of stress concentrations and their distributions at the matrix–particle interface and within the particle. Some results different from the scale-independent case are obtained as: (1) the maximum stress concentration factor (SCF) at the particle–matrix interface is dramatically increased by the size effect especially for the slender particle. This is likely to trigger the void nucleation at the matrix–particle interface by cleavage or atomic separation. (2) At a given overall effective strain, the particle size effect significantly elevates the stress level at the matrix–particle interface. This means that the size effect is likely to advance the interface separation at a smaller overall strain. (3) For scale-independent cases, the elongated particle fracture usually takes place before the interface debonding occurs. For scale-dependent cases, although the SCF within the particle is also accentuated by the particle size effect, the SCF at the interface rises at a much faster rate. It indicates that the probability of void nucleation by the interface separation would increase.

Introduction

The second-phase particles or fibers are often embedded into ductile materials to improve their mechanical properties. However, the higher stress concentration at the particle–matrix interface or within the particle induced by the second phase hard particle or fiber may result in the particle–matrix interface separation (Keer et al., 1973) or the particle fracture (Gurland and Plateau, 1963, Lloyd, 1991). The former failure is generally governed by the traction across the interface (Needleman, 1987) and the interface debonding strength, while the latter is related to the opening stress within the particle and the particle fracture strength (Brechet et al., 1991). If the particle and interfacial strengths are known a priori, a rational void nucleation prediction through an accurate determination of the stress distributions within the particle and at the interface is possible. Several studies on these stress distributions at the interface and within the particle have been performed to explain the void nucleation mechanism (Thomson and Hancock, 1984, Wilner, 1988, Wilner, 1995, Tvergaard, 1993, Tvergaard, 1995). Various possible influences, which come from the particle morphology such as the particle shape (Lee and Mear, 1999); from the particle distribution (Brockenbrough et al., 1992, Ganguly and Pool, 2004); from the material parameters such as the elastic constants (including Young’s modulus and Poisson’s ratio), the yield stress and the hardening exponent (Christman et al., 1989, Bao et al., 1991, Tvergaard, 1990a); and from interface features such as the perfect bonding and partial debonding (Tvergaard, 1990b, Tvergaard, 1993, Tvergaard, 1995), on the mesoscopic and the macroscopic response of MMC have been addressed carefully. These works contributed to a good understanding of the size-independent damage initiation in MMC, but they cannot capture the size effect at the micron or submicron scale.

Recent experiments and computational simulations have repeatedly demonstrated that composites reinforced by second-phase particles display strong size effects when the main size of particles is in the micron or submicron size range. Lloyd (1994) observed that a decrease of the particle size can markedly increase the stiffness of SiC-reinforced aluminum for given material parameters and particle volume fraction. By the transmission electron microscope (TEM) technique, Barlow and co-workers (Barlow and Hansen, 1991, Barlow and Hansen, 1995, Barlow and Liu, 1998) continually observed that the strain gradient distribution around the micron scale whiskers is much smoother than that predicted by the size-independent FEM. Recent simulations based on the discrete dislocation plasticity also validated that the stress level and distribution around the reinforcement strongly depend on the particle size (Cleveringa et al., 1997, Cleveringa et al., 1999a, Khraishi et al., 2004). Similar size effect is also found in other experiments and theoritical analyses such as the micro-indentation (Ma and Clark, 1995, Mcelhaney et al., 1998, Rashid and Voyiadjis, 2004), the micro-twin (Fleck et al., 1994), the micro-bend (Stolken and Evans, 1998, Wang et al., 2003, Cleveringa et al., 1999b) and the metallic materials containing microvoids (Shu, 1998, Khraishi and Khaleel, 2001, Taylor et al., 2002). The classical local plasticity theories fail to explain this size effect due to the lack of material intrinsic length. In recent years, various high-order and lower-order non-local theories possessing the intrinsic length have been developed to capture the size-dependent behavior of materials (Huang et al., 2004, Hwang et al., 2004, Li et al., 2003, Li and Huang, 2005). Although the presently available constitutive models are far from being firmly established (Hutchinson, 2000, Gudmundson, 2004), many investigators actively employed them to probe into the size-dependent response of the particle-reinforced composites; see Fleck and Hutchinson, 1993, Fleck and Hutchinson, 1997, Dai et al., 1999, Shu and Barlow, 2000, Huang et al., 2000, Xue et al., 2002, Niordson and Tvergaard, 2001, Niordson and Tvergaard, 2002, Niordson, 2003, Bittencourt et al., 2003, for example. A general conclusion is reached: smaller particles tend to dramatically elevate the bulk stiffness of MMC and the microscopic stress within matrix and particles.

Previous works on the particle size effect include two main limitations. First, the influence of particle shape is not carefully taken into account. In fact, the particles dispersed in the matrix rarely have idealized shapes like spheres or cylinders. Significant strain gradients inevitably develop at the high curvature regions of non-spherical particles. Second, the bulk constitutive response of the particle-reinforced composites are paid sufficient attention whereas details about stress distributions within particles and matrix are usually neglected, although they are crucial to understand the scale-dependent void nucleation mechanism. So far, it is not clear how particle shape and size jointly influence the stress distribution and the void nucleation mechanism when the particle is in the micron or submicron size range.

Motivated by these backgrounds, we perform here detailed investigations on the coupled effects of particle size and shape on the stress concentration and void nucleation mechanism. A boundary value problem of an infinite matrix containing an isolated prolate spheroidal particle has been theoretically analyzed and numerically solved by a Ritz procedure. For the sake of simplicity, our attention is restricted to the remote proportional and monotonic axisymmetric small deformation tension loading.