PSI - Issue 23

Zdeněk Machů et al. / Procedia Structural Integrity 23 (2019) 535 – 540 Zdeněk Machů / Structural Integrity Procedia 00 ( 2019) 000 – 000

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2.2. Calculation of bending stresses σ appl (z) induced by a kinematic excitation To calculate bending stresses σ appl ( z ) induced by a kinematic excitation of the considered multilayer beam upon steady-state vibrations, an analytical model introduced in Machu et al. (2018) will be employed. This analytical model requires both that the considered beam satisfies assumptions of the Euler-Bernoulli beam theory and that the collecting electrodes of piezo-layers are present over the whole length of the beam. During vibrations, the beam is subjected purely to bending. The steady-state response of the considered beam is obtained by solving the following PDE - Machu et al. (2018): 5 4 4 4 * * 2 1 2 * ( , ) 2 ( , ) ( , ) 0 r y y b w x t w x t w x t J J m x x t t            , (2) i i e Ed  denotes piezoelectric modulus of the i -th layer, where d 31,i is a piezoelectric charge coefficient for 31 -mode of the i -th layer (i.e. polarization direction 3 is considered in the z -axis and elongation of the beam is considered in the x -axis – induced by bending of the beam – see Fig. 1), and 33, S i  denotes the permittivity of the i -th layer measured upon constant mechanical strain. b r is the damping ratio, 1 * N i i i m B h     is the mass of a composite beam per unit of its length and Ω 1 is the first undamped angular natural frequency of the considered beam. The amplitude of stresses in individual layers is calculated using the following expression fromMachu et al. (2018). The calculated stresses are in general of complex values (due to damping present in the structure). However, only the amplitude of induced mechanical stresses is needed for a further assessment and is given by the following relation (vertical bars – | | – in relation (3) represent the modulus of a complex number):     2 2 31, 2 2 33 2 2 31, , 33, 0 0 ( ) 1 i i Ti Ti i i x x i appl i S S x L e i e z w z E z z z B B L x i w w x x R h                     . (3) The distribution of mechanical stresses | σ appl ( z )| needs to be subsequently transformed into a stress distribution, which resembles a stress state present upon beam bending, as shown in Fig. 2. The tensile stresses within the beam's cross section are then responsible for propagation of surface cracks inside the laminate. where      31, 2 1 i S e    31, i S e * J B E z z  3 3 3 3 2 z h 1 1 33, 33, 2 1 1 3 3 y i i i i Ti i i N  i i i z z                  denotes the bending stiffness of a multilayer beam with piezoelectric layers. 31, 31, i

Fig. 2. Transformation of the absolute stress distribution into a bending stress distribution.

2.3. Calculation of the effective R-Curve and K appl using the weight function method

As shown in Majer et al. (2018), the apparent fracture toughness of a multilayer ceramic structure can be represented by the so-called effective R-Curve defined as     , ,0 R eff c res K a K K a   , (4)