Issue 42

J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 283   1 4 4 ; ( ), 0 V H          (18) Problem (VP): Find 2 4 ( , ) u inV V   such that: 0 2 1 0 4 4 ( , ) ( , ) ( , ) ( , ) a u v b v t v ds v V c d u w ds V                           (19) where ( , ) ( ) ( ) ijkl kl ij a u v a e u e v dv    (20) ( , ) i ijk k j v b v e dv x x          (21) ( , ) ij j i c dv x x             (22) ( , ) k ikl l i u d u e dv x x           (23) Proposition1. Problem (VP) is equivalent to Eqs. (7) to (12). Proof. (19) 1 is obtained by multiplying (11) par a test function v i and by integrating by parts; taking into account the boundary conditions (7) and (8). By analogy, we obtain (19) 2 , by multiplying (12) par a test function  and by integrating by parts; taking into account the boundary conditions (9) and (10). The coefficients ijkl a are assumed to be continuous on S  . For the existence and uniqueness of the solution of problem (VP), see [9]. V ARIATIONAL FORMULATION FOR THE PROBLEM OF A FISSURED PIEZOELECTRIC STRUCTURE e now consider a piezoelectric structure containing a closed crack C, i.e. C C  (24) where C is the closure of C, and where C  is assumed to be smooth. Let us introduce the open subset , C  verifying: C C     (25) The local equations of linear piezoelectricity for a fissured piezoelectric structure can then be written as follows [10]: 0 ij C j in x      (26) 0 i C i D in x     (27) W