A three-dimensional, higher-order, elasticity-based micromechanics model

Abstract

The three-dimensional (3D) version of a new homogenization theory [A Two-Dimensional, Higher-Order, Elasticity-Based Micromechanics Model, in print] is presented. The 3D theory utilizes a higher-order, elasticity-based cell model (ECM) analysis for a periodic array of 3D unit cells. The unit cell is discretized into eight subregions or subcells. The displacement field within each subcell is approximated by a (truncated) eigenfunction expansion of up to fifth order. The governing equations are developed by satisfying the pointwise governing equations of geometrically linear continuum mechanics exactly up through the given order of the subcell displacement fields. The specified governing equations are valid for any type of constitutive model used to describe the behavior of the material in a subcell. The specialization of the theory to lower orders and to two-dimeinsions (2D) and to the exact one-dimensional (1D) theory is discussed. Since the proposed 3D homogenization theory correctly reduces to both 2D and 1D the validation process applied to the 2D theory [A Two-Dimensional, Higher-Order, Elasticity-Based Micromechanics Model, in print] directly applies to the current formulation. Additional comparisons of the predicted responses obtained from the 3D ECM theory with existing published results are conducted. The good agreement obtained shows that the current theory represents a viable 3D homogenization tool. The improved agreement between the current theory results and published results as compared to the comparison of the MOC results and the published results is due to the correct incorporation of the coupling effects between the local fields. Additional results showing the convergence behavior of the average fields as a function of the order of the analysis is presented. These results show that the 1st order theory may not accurately predict the local averages but that consistent and converged behavior is obtained using the higher order ECM theories. The proposed theory represents the necessary theoretical foundations for the development of exact homogenization solutions of generalized, three-dimensional microstructures.