Issue 42

J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 291 So, by virtue of (3)-(5), we can formulate the variation of W and W * as follows: * * * * * * * * * * * * * 1 1 1 2 2 1 1 2 2 1 1 1 2 2 1 2 ijkl ij kl ijkl ij kl ijk i jk Y Y Y ijk i jk ijk i jk Y Y ijkl ij kl ijkl ij kl ijk i ij Y Y Y kl kl W a h h dy a h h dy e h h dy Y Y Y e h h dy e h h dy Y Y a h h dy a h H dy e h h dy Y Y Y H                                (84) * * * * * * * * * * * * * 1 1 1 2 2 1 1 2 2 1 1 1 2 2 2 ij i j ij i j ikl ik l Y Y Y ikl ik l ikl ik l Y Y ij i j ikl ik l j j Y Y W h h dy h h dy e h h dy Y Y Y e h h dy e h h dy Y Y h h dy e h h dy D h Y Y                                 (85) Taking (76) and (77) into account, we get: 1 2 1 2 ij kl i i W H W D H               (86) C ONCLUSION rom the variational formulation of the three-dimensional problem of Linear Piezoelectricity, we deduced that corresponding to a cracked piezoelectric structure. Considering afterward the case of a structure presenting a periodic distribution of cracks, we managed to build, on the homogenization period, the homogenized formulation of the corresponding problem, as a result of an asymptotic development of the solution. A non- linear law between the mechanical strain and the electric potential on one hand, and the mechanical stress and the electric displacement on the other hand, has been then established. R EFERENCES [1] Dieulesaint, E., Royer, D., Ondes élastiques dans les solides. Application au traitement du signal, Paris, (1974). [2] Alshits, I., Darinskii, A.N., Lothe, J., On the existence of surface waves in half-anisotropic elastic media with piezoelectric and piezomagnetic properties, Wave. Motion., 16 (1992) 265 – 283. [3] Li, J.Y., Dunn, M.L., Micromechanics of magnetoelectroelastic composite materials: average fields and effective behavior, J. Intell. Mater. Syst. Struct., 7 (1998) 404 – 416. [4] Zhag, T.Y., Tong, P., Fracture mechanics for a mode III crack in a piezoelectric material, Int. J. Solids. Struct, 33 (1996) 343 – 359. [5] Suo, Z., Kuo, C.M., Barnett, D.M., Willis, J.R., Fracture mechanics for piezoelectric ceramics, J. Mech. Phys. Solids, 40 (1992) 739 – 765. [6] Gao, C.F., Wang, M.Z., Periodical cracks in piezoelectric media, Mech. Rech. Commun, 26 (1999) 427 – 432. F