Issue 29

C. Maruccio et alii, Frattura ed Integrità Strutturale, 29 (2014) 49-60; DOI: 10.3221/IGF-ESIS.29.06 52 stresses and strains around a certain macroscopic point can be found by averaging microstructural stresses and strains in a small representative area of the microstructure attributed to that point. This allows finding a globally homogeneous medium equivalent to the original composite, where the equivalence is intended in an energetic sense as per Hill’s balance condition [12,15,16]. Formulated for the electromechanical problem, Hill’s criterion in differential form reads: 1 1          ij ij i i ij ij i i T S D E T S dV D E dV V V (6) and requires that the macroscopic volume average of the variation of work performed on the RVE is equal to the local variation of work on the macroscale. In the previous equation: ij T , ij S , i D and i E represent respectively the average values of stress, strain, electric displacement and electric field components and V is the RVE volume. Hill’s lemma leads to the following equations: 1 1 1 1         ij ij ij ij i i i i T T dV S S dV V V D D dV E E dV V V (7) Fig. 2 shows a two-dimensional RVE. Periodic boundary conditions imply that, on two opposite edges, the displacement and the electric potential are equal, the stress vector and the electric displacement vector are opposite. In the implementation, these constraints are enforced by prescribing the primal variables at the corner nodes and using the Lagrange multiplier method. Figure 2 : Enforcing periodic boundary conditions During the initial periodicity of the RVE, for every respective pair of nodes on the top–bottom ( 4 3 1 2 ,   X X ) and right– left ( 1 4 2 3 ,   X X ) boundaries this relation is valid in the reference configuration: 4 3 1 2 4 1      X X X X 1 4 2 3 2 1      X X X X (8) where p X , p = 1, 2, 4 are the position vectors of the corner nodes 1, 2, and 4 in the undeformed state. Then in the deformed configuration the previous relations lead to   4 3 1 2 4 1      M x x F X X   1 4 2 3 2 1      M x x F X X (9) where M F is the deformation gradient. Now if the position vectors of the corner nodes in the deformed state are prescribed according to:  p M p x F X with p = 1, 2, 4 (10) then the periodic boundary conditions may be rewritten in terms of displacement and electric potential as: