PSI - Issue 23

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

539

5

where K c,0 denotes the intrinsic fracture toughness of a current layer and K res ( a ) is the stress intensity factor caused by thermal residual stresses σ res ( z ), which is calculated using the weight function method as follows:       0 , a res res K a h z a z dz    . (5) In this relation, h ( z , a ) is a weight function defined in Fett and Munz (1992). Similarly, the SIF caused by the applied bending stress, which is induced by a kinematic excitation, can be calculated as       0 , a appl appl K a h z a z dz    . (6) 2.4. Determination of the maximal allowable load In our case, the maximal allowable load is such an amplitude of the beam’s clamped end acceleration ü 0 , which satisfies the following relation     , 0 , R eff crit appl crit K K a K a ü k  , (8) where a crit is the critical crack length, which is in our case represented by the thickness of protective layers, and k K is a safety factor, which was set to be of value 1.5. 3. Results An analytical model of the piezoelectric harvester in the form of a multilayer beam, which is illustrated in Fig. 1, was used to calculate the maximal allowable acceleration of the beam’s clamped end for various forcing frequencies that are close to the beam’s first natural frequency . Dimensions of the used beam were L x B x W =80x10x1 mm and the laminate was composed of ZrO 2 , ATZ and BaTiO 3 based materials as also shown in Fig. 1. The beam had a tip mass M t =10 g, which shifts its first natural frequency to 49.85 Hz. The damping ratio b r is considered to be 0.01. The Fig. 3(a) shows the calculated effective R- Curve for Δ T = – 1430 °C (from section 2.1). An unstable crack growth initiates, if following conditions from Kotoul et al. (2012) are met:     , appl R eff K a K a  , where     , appl R eff dK dK a a da da  . (7)

a)

b) )

max

ü 0 ü 0

(49.85 Hz) = 0.88 g crit (49.85 Hz) = 1.32 g

K R,eff K appl ( ü 0 K appl ( ü 0

max ) crit )

K appl , K R,eff [MPa·m 1/2 ]

11.7

1 st natural frequency 49.85 Hz

7.8

Fig. 3. (a) The calculated R-Curve of the multilayer structure K R,eff together with the stress intensity factors K appl at the tip of a crack propagating from the surface inside the laminate, corresponding to load conditions upon vibrations at the first natural frequency; (b) maximal allowable acceleration applied on the clamped end of the considered harvester as a function of the forcing frequency.