| A homogenization theory is developed to determine the overall elastoplastic behavior of a particle-reinforced, fiber-reinforced composite with a ductile interphase. Unlike most existing homogenization theories which are primarily concerned with the ordinary two-phase composites, the present one is confronted with two ductile phases, with one enclosing the other. The theory is developed with an energy approach in conjunction with a field-fluctuation method for the calculation of the homogenized effective stress of the ductile phases. To offer an alternative to the energy approach, a new mean effective stress approach is also introduced. These two approaches are compared in light of an exact elastic-plastic analysis, and it is found that both approaches could provide plausible estimates for the overall behavior of the composite. The prediction by the energy approach, however, is slightly more accurate. The developed theories are applicable to the 3-phase system regardless whether the interphase is more ductile or stiffer than the matrix. When the interphase is more ductile, it is shown that even the presence of a thin layer can have a very significant effect on the plasticity of the overall composite.;In the context of a functionally-graded composite whose matrix property varies linearly from the interface, exact analytical solutions are derived for the effective bulk modulus of a particle-reinforced composite and the plane-strain bulk modulus of a fiber-reinforced composite. Unlike the replacement method or the series solution, a unique feature of the results here is that the effective properties are given explicitly, rendering them useful for future applications. Specifically four types of variation are considered for each class of composites. In the particle case (i) the bulk modulus of the matrix varies linearly while its shear modulus remains constant, (ii) the shear modulus varies linearly while its bulk modulus remains constant, (iii) both the bulk and the shear moduli vary linearly but their extrapolated values to the origin are zero, and (iv) incompressible matrix. In the fiber case it is the plane-strain bulk modulus of the matrix instead of the ordinary 3-D bulk modulus that varies. Some calculations are presented to illustrate the influence of the slope of the modulus variation on the overall properties of the composites. |
