Issue 29

C. Maruccio et alii, Frattura ed Integrità Strutturale, 29 (2014) 49-60; DOI: 10.3221/IGF-ESIS.29.06 51 potential in case of closed contact. The electromechanical constraints are regularized with the penalty method. For each slave node, the normal gap is computed as: ( )    s m N g x x n (2) where s x is the position vector of the slave node, m x is the position vector of its normal (i.e. minimum distance) projection point onto the master surface, and n is the outer normal to the master surface at the projection point. The sign of the measured gap is used to discriminate between active and inactive contact conditions, a negative value of the gap leading to active contact. The electric field requires the definition of the contact electric potential jump: ( )      s m g (3) where  s and  m are the electric potential values in the slave node and in its projection point on the master surface. A tensor product representation of one-dimensional Bézier polynomials is used to interpolate the master surfaces in the contact interface for a three-dimensional problem according to the relation:   1 2 1 2 0 0 , ( ) ( )         m m m m kl k l k l x B B d (4) where 1 ( )  m k B and   2  m l B are the Bernstein polynomials and kl d the coordinates of the control points [22]. With the same procedure, to interpolate the electric potential on the master surface the following relation is introduced:   1 2 1 2 0 0 , ( ) ( )           m m m m kl k l k l B B (5) where  kl is the potential evaluated at 16 control points as a function of the potential at the auxiliary points ˆ  kj and at the master nodes  m ij , see Fig.1. a) b) c) Figure 1 : Electromechanical contact elements: a) General master-slave concept b) Node to surface discretization c) Smoothing with Bezier patches According to standard finite element techniques, the global energy of the system is obtained by adding to the variation of the energy potential representing the continuum behaviour the virtual work associated to the electromechanical contact contribution provided by the active contact elements. Performing the first and second variation of the global energy the global set of equations is obtained. When an external load is applied to the RVE, the stress, strain and electric fields in the microstructure will show large gradients due to the microstructural heterogeneity. However, due to the differences in scale, the microstructural electric/deformation field around a macroscopic point will be approximately the same as the electric/deformation field around neighbouring points. The repetitive deformations justify the assumption of local periodicity, meaning that the microstructure can be thought as repeating itself near a macroscopic point. However, the microstructure itself may differ from one macroscopic point to another. The repetitive microstructural generalized deformations suggest that macroscopic