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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In this paper, the apparent fracture toughness of a multilayer structure (depending on the used materials and their thicknesses) is analyzed using a developed analytical model and the maximal allowable acceleration of the considered harvester upon a given forcing frequency (ensuring that no surface cracks can propagate through the structure) is determined. 2. Determination of the resistance to surface crack propagation of a multilayer piezoelectric harvester The considered piezoelectric harvester in the form of a multilayer ceramic cantilever beamwith piezoelectric layers, which is shown in Fig. 1, is kinematically excitated by a time-harmonic function. The beam is excitated by forcing frequencies very close to the beam’s first natural frequency , which means that the beam ’s vibrations consist purely of the first-mode shape upon steady-state vibrations. Therefore, bending stresses are induced in the considered structure with peak values occurring at the beam’s clamped end (upon maximal deflection of the free end). The stress distribution in individual layers at the beam’s clamped end upon steady-state vibrations is denoted as σ appl ( z ). The considered multilayer structure is also affected by thermal residual stresses σ res ( z ), generated upon the laminate processing. The considered laminate, composed primarily of ZrO 2 , ATZ (Alumina Tetragonal Zirconia) and BaTiO 3 ceramic layers (as shown in Fig. 1), is prepared using the electrophoretic deposition (EPD) and sintered at ~1500 °C.

Fig. 1. Scheme of the considered piezoelectric harvester.

Thermal residual stresses in individual layers (induced upon the laminate cooling due to a mismatch in coefficients of thermal expansion of used materials) influence the apparent fracture toughness of such a laminate, which can be represented by the so-called effective R-Curve. Bending stresses induced by the kinematic excitation can be represented by the stress intensity factor K appl . An effective way to compute both these quantities is to employ the weight function method, which was proved to be a very effective and fast tool for this purpose - as can be seen in the previous authors’ work , Majer et al. (2018) or Kotoul et al. (2012). Both these quantities can be used for an estimation of the resistance of a given multilayer configuration to unstable propagation of surface cracks. 2.1. Calculation of thermal residual stresses Thermal residual stresses in individual layers can be calculated using relations coming from the classical laminate theory. The following relation (1) is used to calculate the magnitude of thermal residual stresses in the i -th layer of a laminate loaded purely by a temperature change  T - see Sestakova et al. (2011) :   , 1 1 , where 1 1 1 N N i i i i i i res i i i i i i i E E h Eh T                    . (1) Here, E i , ν i and α i is the elastic modulus , Poisson´s ratio and thermal expansion coefficient of the i -th layer respectively and Δ T is a temperature difference between the actual (room) and the zero-strain (reference) temperature. The zero-strain (reference) temperature for the ceramic materials, mentioned in Fig. 1, is considered to be roughly 1450 °C ( while the sintering temperature is around 1500 °C) . Therefore, for the calculation of residual stresses in the multilayer structure we will consider Δ T = – 1430 °C (difference between the reference and room temperature) - see Sestakova et al. (2011). The term  represents the apparent thermal expansion coefficient of the whole laminate and h i is the thickness of the i -th layer. The distribution of σ res ( z ) is assumed to not be affected by piezoelectric properties of particular layers.