PSI - Issue 80

Akihide Saimoto et al. / Procedia Structural Integrity 80 (2026) 352–367 A. Saimoto et al. / Structural Integrity Procedia 00 (2023) 000–000

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Regarding the problem of electroelastic media with cracks, some important studies relevant to the present research are as follows: Sosa (1991) analyzed piezoelectric materials with elliptical hole using complex potentials. The bound ary condition used by Sosa at the rim of an elliptical hole was based on the so-called impermeable assumption, where the surface forces and the normal component of the electrical displacement at the elliptical boundary were set to zero. As the vacuum has non-zero dielectric constant, ϵ 0 = 8 . 854 × 10 − 12 F / m, such a electrically impermeable treatment is considered as an approximation. Then Sosa and Khutoryansky (1991) and Gao and Fan (1999) extended the analytical condition to account the presence of a vacuum or air inside the hole. As for a half plane problem, a concentrated force or electric charge on the surface of an electroelastic half-plane was studied by Sosa and Castro (1994). Sung and Liou (1995) and Yang et al. (2007) performed a crack analysis in a semi-infinite electroelastic solid. Shindo et al. (1998) performed an electrostatic analysis of half plate with surface electrodes by Fourier transform technique. Ueda et al. (2006) studied a crack problem in a piezoelectric half plane subjected to thermal loading. The Green’s function of piezoelectric half plane was studied by Liu et al. (1997), and Gao and Fan (1998) using Lekhnitskii formalism and Gao and Noda (2004) using Stroh formalism. Khoroshev and Glushchenko (2020) showed how to analyze the problem of a semi-infinite piezoelectric plate with cracks using an infinite series expansion of complex potentials. As for published text books on general electroelastic and magneto electroelastic problems, including one by Qin (2012) and another by Bardzokas et al. (2007). In the present study, complex potentials representing the influence of concentrated forces and electric charges acting in a piezoelectric semi-infinite plate through extended Lekhnitskii formulation is introduced. Then, by di ff er entiating these potentials with respect to the source point, the fundamental solutions of the force and charge doublets are obtained. Then it is shown that the desired crack problem can be expressed as boundary integral equations by appropriately embedding these doublets in the piezoelectric semi-infinite plate along the line or curve where the crack is to be located. The method of analyzing the crack problems used in this paper, known as the body force method (BFM), is based on the principle of superposition and can e ffi ciently handle a variety of crack problems. In particular, the present study performs an analysis of an edge crack in the piezoelectric semi-infinite plate subjected to tension parallel to the free edge at infinity.

Nomenclature

σ i j component of stress, [N / m 2 ] ε i j component of strain, [-]

E i component of electric field, [V / m] or [N / C] D i component of electric displacement, [C / m 2 ] u i component of elastic displacement, [m] ϕ electrostatic potential, [V] s i j elastic compliance, [m 2 / N] d i j piezoelectric constants, [m / V] or [C / N] ϵ i j permittivity constants, [F / m] or [C / Vm] ϵ 0 permittivity of vacuum, [F / m] or [C / Vm] µ i eigenvalue (solution of characteristic equation), [-]

2. Fundamental equations

2.1. 2-dimensional Electroelasticity

In general, a poled piezoceramics can be modeled by a transversely isotropic materials and there are 4 possible di ff erent types of piezoelectric constitutive equations. In the present paper, so-called strain-charge form is employed.

E i jk ℓ σ k ℓ + d ki j E k , D i = d ik ℓ σ k ℓ + ϵ σ ik E k

ε i j = s

(1)