Issue 42

M. Peron et alii, Frattura ed Integrità Strutturale, 42 (2017) 223-230; DOI: 10.3221/IGF-ESIS.42.24 226 library were used, respectively for plane and solid models, and the magnetic field has been introduced by a voltage difference. The coordinate axes x = x 1 and z = x 3 are chosen such that the y = x 2 axis coincides with the thickness direction and such that the easy axis of magnetization is the z-direction. Because of symmetry, only the half of the model was used in the FEA. Before carrying out simulations, a mesh sensitivity study was undertaken to determine the adequate finite element (FE) number to be used. SED value have been first determined from a very refined mesh and then from some coarser meshes. The refined mesh had the same FE number adopted in a previous work by the authors, in which finite element models with 6400 elements were used to evaluate the energy release rate by means of J-integral on the same geometry. Among different coarse mesh patterns, it has been found suitable for compute SED without accuracy lost a mesh with 274 elements, of which at 10 elements placed inside the control volume. The results are summarized in Tab. 1, where the SED value from the proposed coarse mesh is compared with that from the very refined one. The mesh insensitivity is a consequence of the finite element method, in which the elastic strain energy is computed from the nodal displacements, without involving stresses and strains, as shown by Lazzarin et al. [10]. The relationship between magnetostriction and magnetic field intensity is essentially non-linear. Nonlinearity arises from the movement of the magnetic domain walls, as shown by Wan et al. [20]. To take into account this non-linear behavior, the constants d 15 , d 31 and d 33 for Terfenol-D, in presence of B z = B 0 , are given by: 15 15 31 31 31 33 33 33 m m z m z d d d d m H d d m H      (10) where 15 m d , 31 m d and 33 m d are the piezomagnetic constants, whereas m 31 and m 33 are the second order magnetoelastic constants. Jia et al. [21] proved that if the specimen's dimension in the direction in which the magnetic field is applied is at least two times greater than the other two dimensions, then the longitudinal magnetostriction is prevailing and it can be assumed that only d 33 is a function of the magnetic field H z and that m 31 is equal to zero. Number FE (control volume) Control volume Control volume Number FE model) W [MJ.m - 3 ] W  [%] 128 6400 0.01461 - 10 274 0.01457 -0.3 Table 1 : Mean values of SED for different mesh refinement. Material Elastic compliance [10 -12 m 2 .N -1 ] Piezo-magnetic constants [10 -9 mA -1 ] Magnetic permeability [10 -6 Hm -1 ] Densit y [kg.m -3 ] 11 H S 33 H S 44 H S 12 H S 13 H S 31 m d 33 m d 15 m d 11 T  33 T  ρ Terfenol-D 17.9 17.9 26.3 -5.88 -5.88 -5.3 11 28 6.29 6.29 9250 Table 2 : Terfenol-D material properties.