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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The apparent fracture toughness of the laminate, which is in fact the apparent fracture toughness of used protective layers, was calculated to be  11.7 MPa·m 1/2 . Compared to the intrinsic fracture toughness of the bulk ATZ material, which is K Ic ATZ = 3.2 MPa·m 1/2 , a significant improvement was obtained. Then, considering the safety factor k K = 1.5, the maximal allowable stress intensity factor induced upon the mechanical loading is:   max 1/2 0 0.14 mm, 11.7 /1.5 7.8MPa m appl K a u     . (9) Value of K appl from (9) is then used in calculation of the maximal allowable acceleration of the beam’s clamped end as a function of forcing frequency, which is presented in g -multiples in Fig. 3(b). As expected, the minimal value of 0.88 g occurs for the forcing frequency equal to the beam’s first natural frequency. For forcing frequencies receding from the first natural frequency, the values of ü 0 max steeply rise (based upon general expectations). 4. Conclusions In this paper, an analytical model for estimation of surface crack propagation in a multilayer piezoelectric harvester is presented. The model utilizes the classical laminate theory to calculate thermal residual stresses in the particular layers of the harvester and the theory of multilayer beam vibrations (used already in the authors’ previous work ) to calculate mechanical stress components in the structure induced by a kinematic excitation. The weight function method is used to estimate the apparent fracture toughness of the structure and also its resistance to surface crack propagation. Furthermore, the presented model is able to estimate the maximal allowable acceleration (upon various forcing frequencies) of a multilayer piezoelectric harvester (in our case made of ZrO 2 /ATZ/BaTiO 3 ceramic materials) that can still be applied to the beam’s clamped end without propagation of potential surface cracks inside the laminate. Amplitudes of the maximal allowable acceleration with a given safety factor were determined for a range of frequencies in a region close to the first natural frequency (where operation of the harvester is expected). Based upon the results and developed analytical models, the material/layer composition of the energy harvester can be tailored to a given application with a primary aim to enhance the harvester’s power capabilities upon preservation of its integrity at given excitation frequencies. Acknowledgements A financial support through a grant no. 17-08153S of the Czech Science Foundation is gratefully acknowledged. References Erturk, A., Inman, D.J. 2009. An experimentally validated bimorph cantilever model for piezoelectric energy harvesting from base excitations. Smart Materials and Structures 18, 025009. Erturk, A., Inman, D.J., 2011. Piezoelectric Energy Harvesting. 1st (Ed.). John Wiley & Sons, UK, pp. 392. Fett, T., Munz, D., 1992. Determination of fracture toughness at high temperatures after subcritical crack extension. Journal of the American Ceramic Society 75, 3133-3136. Kotoul, M., Sevecek, O., Vyslouzil, T., 2012. Crack growth in ceramic laminates with strong interfaces and large compressive residual stresses. Theoretical and Applied Fracture Mechanics 61, 40-50. Machu, Z., Sevecek, O., Majer, Z., Hadas, Z., Kotoul, M., 2018. Optimization of the electro-mechanical response of the multilayer piezoelectric energy harvester, 2018 18th International Conference on Mechatronics-Mechatronika (ME)IEEE, 1. Majer, Z., Sevecek, O., Machu, Z., Stegnerova, K., Kotoul, M., 2018. Optimization of Design Parameters of Fracture Resistant Piezoelectric Vibration Energy Harvester, Key Engineering Materials 774, 416-422. Phung-Van, P., Lorenzis, L.D., Thai, C.H., Abdel-Wahab, M., Nguyen-Xuan, H., 2015. Analysis of laminated composite plates integrated with piezoelectric sensors and actuators using higher-order shear deformation theory and isogeometric finite elements. Computational Materials Science 96, 495-505. Sestakova, L., Bermejo, R., Chlup, Z., Danzer, R., 2011. Strategies for fracture toughness, strength and reliability optimisation of ceramic-ceramic laminates. International Journal of Materials Research 102, 613-626. Toudehdehghan, A., Rahman, M.M., Nagi, F., 2017. Dynamic analysis of composite beam with piezoelectric layers under thermo-mechanical load. IOP Conference Series: Materials Science and Engineering 257. Zielinski, M., Mieyeville, F., Navarro, D., Bareille, O., 2014. A low power wireless sensor node with vibration sensing and energy harvesting capability, 2014 Federated Conference on Computer Science and Information Systems, 1065-1071.