Acta Mechanica 121, 165-176 (1997) ACTA MECHANICA 9 Springer-Verlag 1997 Strain gradients and continuum modeling of size effect in metal matrix composites H. To Zhu and H. M. Zbib, Pullman, Washington, and E. C. Aifantis, Houghton, Michigan (Received June 4, 1995; revised December 27, 1995) Summary. Constitutive modeling for the particle size effect on the strength of particulate-reinforced metal matrix composites is investigated. The approach is based on a gradient-dependent theory of plasticity that incorporates strain gradients into the expression of the flow stress of matrix materials, and a finite unit cell technique that is used to calculate the overall flow properties of composites. It is shown that the strain gradient term introduces a spatial length scale in the constitutive equations for composites, and the dependence of the flow stress on the particle size/spacing can be obtained. Moreover, a nondimensional analysis along with the numerical result yields an explicit relation for the strain gradient coefficient in terms of particle size, strain, and yield stress. Typical results for aluminum matrix composites with ellipsoidal particles are calculated and compare well with data measured experimentally. 1 Introduction Experimental results have shown that the plastic behavior of particulate-reinforced metal matrix composites (MMCs) is influenced significantly by the particle size/spacing (for example, Kamat et al. [1], [2]). One way to model this size/spacing effect is to use dislocation theoris. For example, the Orowan model has been applied to small particles (Brown [3]) and predicted that the flow stress is proportional to the inverse of the particle spacing, while in the Ashby model (Ashby [4]) this dependence is the inverse of the square root of the particle spacing. Recently, a bowed-out tilt-wall model based on the dislocation pile-up was proposed (Rhee et al. [5], [6]) where the flow stress is proportional to L -~ (L is the edge-edge particle spacing) together with a weak logarithmic dependence on L. Alternatively, this issue may be addressed within the framework of continuum mechanics. We mention, for example, the recent contributions of Li and Weng [7], Ju and Tseng [8], Ponte Castaneda [9], Lee and Mear [10], Bat et al. [11], and Zbib and Zhi [12], who dealt with the effective nonlinear properties of MMCs. However, when applying classical phenomenological constitutive theories to MMCs, they have difficulties describing the size dependence. This is because these theories, originally developed for conventional metals or metallic alloys, assume that the deformation field is homogeneous, and thus do not include a deformation length scale in the constitutive equations. In fact, classical continuum mechanics is local in character in that the stress at a material point is considered to be a functional of past deformation history of that point only. Eringen and co-workers have extended classical continuum mechanics (see, for example, Eringen, [13] - [15]) to include nonlocal effects. The basic assumption of nonlocal continuum mechanics is that the stress at a point is a functional of the past deformation histories of all materials points of the body, resulting in a spatial integral form of constitutive equations. Although difficult to use (especially when nonlinearities and finite domains are involved), the nonlocal theory is