PSI - Issue 80

Miroslav Hrstka et al. / Procedia Structural Integrity 80 (2026) 471–492 M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000 – 000

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Table 1. Elastic, electrical and thermal properties PZT-5H ( Material 1 ) poled in x 1 -direction

SiO 2 ( Material 2 )

Elastic properties 11 E C x10 10 [Pa] 12 E C x10 10 [Pa] 23 E C x10 10 [Pa] 22 E C x10 10 [Pa] 44 E C x10 10 [Pa] Electrical properties 11 e [Cm -2 ]

11.7

7.1

5.3 5.5

1.45 1.45

12.6 3.53

7.1

(C

11 -C 12 )/2=2.8

23.3 -6.5 17.0 13.0 15.1 800 0.39

0 0 0

[Cm [Cm

-2 ] -2 ]

12 e 26 e

11   x10 -9 [C(Vm) -1 ] 22   x10 -9 [C(Vm) -1 ]

0.035 0.035

E c [kVm

-1 ]

- - -

P s [N(Vm)

-1

s  [-]

0.004

Thermal properties α 11 x10 -6 [°C -1 ]

1.2 1.2 1.2

0.56 0.56

α 22 x10 α 33 x10

-6 [°C -1 ] -6 [°C -1 ]

0.56 Firstly, dependence of the singularity exponents δ on the notch angle ω 1 for the studied bi-material combination with PZT-5H poled in the x 2 direction ( α 1 = 90°) is shown in Fig. 2. The graphs have the typical character of the ε type piezoelectric bi-materials described in (Hrstka, Profant, & Kotoul, 2019; Ou & Chen, 2004). Nevertheless, there are some dissimilarities. Firstly, the roots δ ͳ ǡ δ ʹ approach the limit value 1, respectively 2 asymptotically, but δ ͵ remains almost constant in the whole investigated range. The close proximity to the-pole in 1.0 brings about numerical troubles in the root finding algorithm mpmath package findroot which has to be set properly. Additionally, in the latter bi-material combination there is a region between 21° and 26° where root δ ͳ ‹• ‘’Ž‡šǤ –• complex conjugate counterpart is not depicted in the graph as it is the non-singular term. The region where δ ͳ is complex conjugate is wider and the imaginary parts of the roots do not reach their maximal value for ω 1 = 180 ° , as was typical for pure piezoelectric bi-materials. In the text, two poling direction orientations were studied, i.e. case 1: α 1 = 90° and case 2: α 1 = 0°. It was shown in Hrstka (2019) that for interfacial cracks with one non-piezoelectric material, the singularity exponents do not vary with poling direction and for the PZT-5H/SiO 2 they are δ 1,2 ൌ 0.5±0.0304i ǡ δ 3 ൌ 0.5. The full field solution u FEM and t FEM was extracted from the finite element model created by the FEniCS 2019 software. The geometry of the sample is shown in Fig. 3a with dimensions a 1 = 10 mm, a 2 = 10 mm, b 1 = 25 mm, b 2 = 50 mm, crack face angles ω 1 = 180 °, ω 2 = – 180 °. The analysis was pure static structural with thermal strains introduced directly into the strain tensors in the generalized Hooke’s law (10). The problem was set as mixed piezoelectric with triangular elements with quadratic base functions. The finite element mesh is shown in Fig. 3b, with domains for defining the material domains and a circular region for domain integral in Eq. (39). These integrals were computed by using 7 point Gauss quadrature over the element extracted from the finite element mesh. There were 132835 nodes and 268698 elements in total. Boundary conditions are depicted in Fig. 3a. As the problem is considered as a study of temperature effect on thin piezoelectric layers which have also practical applications and have been widely studied, the substrate was clamped in the normal direction and on the bottom face zero electric potential was set. The