On Boundary Condition Implementation Via Variational Principles in Elasticity-Based Homogenization

Convergence characteristics of the locally exact homogenization theory for periodic materials, first proposed by Drago and Pindera (2008, “A Locally-Exact Homogenization Theory for Periodic Microstructures With Isotropic Phases,” ASME J. Appl. Mech., 75(5), p. 051010) and recently generalized by Wang and Pindera (“Locally-Exact Homogenization Theory for Transversely Isotropic Unidirectional Composites,” Mech. Res. Commun. (in press); 2016, “Locally-Exact Homogenization of Unidirectional Composites With Coated or Hollow Reinforcement,” Mater. Des., 93, pp. 514–528; and 2016, “Locally Exact Homogenization of Unidirectional Composites With Cylindrically Orthotropic Fibers,” ASME J. Appl. Mech., 83(7), p. 071010), are examined vis-a-vis the manner of implementing periodic boundary conditions. The locally exact theory separates the unit cell problem into interior and exterior problems, with the separable interior problem solved exactly in cylindrical coordinates and the inseparable exterior problem tackled using a balanced variational principle. This variational principle leads to exceptionally fast and well-behaved convergence of the Fourier series coefficients in the displacement field representation of the unit cell's different phases. Herein, we compare the solution's convergence behavior based on the balanced variational principle with that based on the constrained energy-based principle originally proposed by Jirousek (1978, “Basis for Development of Large Finite Elements Locally Satisfying All Fields Equations,” Comput. Methods Appl. Mech. Eng., 14, pp. 65–92) in the context of locally exact finite-element analysis. The relevance of this comparison lies in the recently rediscovered implementation of Jirousek's constrained variational principle in the homogenization of periodic materials.