PSI - Issue 23

Miroslav Hrstka et al. / Procedia Structural Integrity 23 (2019) 419–424 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

424

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3. Numerical results Consider a PZT-5H/BaTiO 3 bi-material notch with the local geometry and free/clamped boundary. The notch geometry is defined by the angles ω 1 = 150 ∘ and ω 2 = −180 ∘ . The poling direction is parallel with 2 -axis, i.e. α 1 = α 2 = 90 ∘ . The phase portrait of the transcendental function (10) is depicted in Fig. 3. It can be observed that on the interval 0 < ℜ { δ } < 1 there are two real and two pairs of complex conjugate roots: δ 1 = 0.2187 , δ 2,3 = 0.2961 ± 0.05114 , δ 4,5 = 0.7883 ± 0.04532 and δ 6 = 0.9079 . The character of the singularity exponents is considerably different in comparison to the case of traction free notch faces. The Fig. 4 shows the exponents for varying value ω 1 and for piezoelectric bi-materials PZT-5H/BaTiO 3 and PZT 5H/PZT-4. One can deduce that the character of the roots δ as a function of the notch geometry is more complicated than for the stress free/free bi-material notches. The asymptotic stresses, electric displacements, displacements and electric potentials calculated along the circular path with radius = 2mm encircling the notch tip in the bi-material PZT-5H/BaTiO 3 together with results obtained by FEM are shown in Fig. 5. The plots show a very good agreement of the asymptotic solution (solid lines) with the complete FEM solution obtained using a very fine mesh.

4. Conclusions

The asymptotic solution of the clamped piezoelectric bi-material notch was derived. It was shown that the singular part of the asymptotic solutions is composed of six components contains two real and two pairs of complex conjugate singularity exponents. The corresponding generalized stress intensity factors were evaluated using the generalized Ψ integral method. The asymptotic solution composted of all six singular components was compared with finite element solution in very good agreement.

Acknowledgements

The support of the Czech Science Foundation through a grant number 17-08153S is gratefully acknowledged.

References

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