This article investigates the dynamic non-local stress analysis of two collinear semi-permeable mode-I cracks in a piezoelectric medium under the harmonic waves by using the generalized Almansi theorem and the Schmidt method. Based on the Fourier transform technique, this problem is formulated into coupled dual integral equations. The dynamic stress and the dynamic electric displacement fields at the crack tips are obtained by solving the derived dual integral equations. Numerical examples are provided to show the effects of the crack length, the distance between the two collinear cracks, the lattice parameter, the electric permittivity of the air inside the crack, and the characteristics of the harmonic wave on the dynamic stress field and the dynamic electric displacement field near the crack tips. The dynamic stress and electric displacement decrease with increasing the distance between two collinear cracks and lattice parameter in a piezoelectric medium. Meanwhile, the dynamic field will impede or enhance crack propagation in piezoelectric medium depending on the circular frequency of the incident wave. Different from the classical solutions, the present solutions exhibit no stress and electric displacement singularities at the crack tips. This work is expected to be helpful for theoretical modeling of piezoelectric medium at nanoscale.
BhargavaRRJangidK (2013) Strip-saturation model for piezoelectric plane weakened by two collinear cracks with coalesced interior zones. Applied Mathematical Modelling37: 4093–4102.
2.
BhargavaRRVermaPR (2016) Strip-electro-mechanical yield model for transversely situated two semi-permeable collinear cracks in piezoelectric strip. Theoretical and Applied Fracture Mechanics81: 32–49.
3.
ChenHSWeiWYLiuJX, et al. (2014) Propagation of a semi-infinite conducting crack in piezoelectric materials: mode-I problem. Journal of the Mechanics and Physics of Solids68: 77–92.
4.
ChenWQLeeKYDingHJ (2004) General solution for transversely isotropic magneto-electro-thermo-elasticity and the potential theory method. International Journal of Engineering Science42: 1361–1379.
5.
DeegWF (1980) The analysis of dislocation, crack and inclusion problems in piezoelectric solids. PhD Thesis, Stanford University, Stanford, CA.
6.
DingHJChenBLiangJ (1996) General solutions for coupled equations for piezoelectric media. International Journal of Solids and Structures33: 2283–2296.
7.
Ebrahimi-NejadSBoreiryM (2018) Comprehensive nonlocal analysis of piezoelectric nanobeams with surface effects in bending, buckling and vibrations under magneto-electro-thermo-mechanical loading. Materials Research Express5: 035028.
8.
ErdelyiA (1954) Tables of Integral Transforms, vol. 1. New York; Toronto, ON, Canada; London: McGraw-Hill.
9.
EringenAC (1976) Non-local polar field theory. In: EringenAC (ed.) Continuum Physics, vol. 4. New York: Academic Press, pp. 205–267.
10.
EringenAC (1978) Line crack subject to shear. International Journal of Fracture14: 367–379.
11.
EringenAC (1979) Line crack subject to anti-plane shear. Engineering Fracture Mechanics12: 211–219.
12.
EringenAC (1983) Interaction of a dislocation with a crack. Journal of Applied Physics54: 6811–6817.
13.
EringenACSpezialeCGKimBS (1977) Crack tip problem in non-local elasticity. Journal of the Mechanics and Physics of Solids25(5): 339–355.
14.
GradshteynISRyzhikIM (1980) Table of Integral, Series and Products. New York: Academic Press.
15.
HaoTHShenZY (1994) A new electric boundary condition of electric fracture mechanics and its applications. Engineering Fracture Mechanics47: 793–802.
16.
JamiaNEl-BorgiSUsmanS (2016) Non-local behavior of two collinear mixed-mode limited-permeable cracks in a functionally graded piezoelectric medium. Applied Mathematical Modelling40: 5988–6005.
17.
KeLLWangYSWangZD (2012) Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory. Composite Structures94: 2038–2047.
18.
LapustaYKomarovALabesse-JiedF, et al. (2011) Limited permeable crack moving along the interface of a piezoelectric bi-material. European Journal of Mechanics A: Solids30: 639–649.
19.
LiuHT (2017) Dynamic non-local theory solution to a permeable mode-I crack in a piezoelectric medium. Engineering Fracture Mechanics179: 43–58.
20.
LiuJZhangPCLinG, et al. (2016) Elastodynamic response for the multi-layered transversely isotropic piezoelectric solid subjected to time-harmonic loads. Composite Structures153: 755–772.
21.
MonfaredMMAyatollahiM (2017) Interactions of multiple cracks in a transversely isotropic piezoelectric plane under mixed mode condition. Engineering Fracture Mechanics180: 87–104.
22.
MorsePMFeshbachH (1953) Methods of Theoretical Physics (Part). New York: McGraw-Hill, pp. 828–930.
23.
NowinskiJL (1984) On non-local aspects of the propagation of Love waves. International Journal of Engineering Science22: 383–392.
24.
PartonVZ (1976) Fracture mechanics of piezoelectric materials. Acta Astronautica3: 671–683.
25.
SharmaKBuiTQBhargavacRR, et al. (2016) Numerical studies of an array of equidistant semi-permeable inclined cracks in 2-D piezoelectric strip using distributed dislocation method. International Journal of Solids and Structures80: 137–145.
26.
ViunOKomarovALapustaY, et al. (2018) A polling direction influence on fracture parameters of a limited permeable interface crack in a piezoelectric bi-material. Engineering Fracture Mechanics191: 143–152.
27.
YangFQ (2001) Fracture mechanics for a mode I crack in piezoelectric materials. International Journal of Solids and Structures38: 3813–3830.
28.
YangJS (2006) Analysis of Piezoelectric Devices. Hackensack, NJ: World Scientific Publishing Co Pte Ltd.
29.
YauWF (1967) Axisymmetric slipless indentation of an infinite elastic cylinder. SIAM Journal on Applied Mathematics15: 219–227.
30.
YuHJWangJKozinovS, et al. (2018) Phase field analysis of crack tip parameters in ferroelectric polycrystals under large-scale switching. Acta Materialia154(1): 334–342.
31.
ZhangWXWangH (2018) Axisymmetric boundary condition problems for transversely isotropic piezoelectric materials. Mechanics Research Communications87: 7–12.
32.
ZhaoMHLiYFanCY (2018) Effects of thermal and electric boundary conditions on fracture of three-dimensional thermopiezoelectric media. Journal of Intelligent Material Systems and Structures131: 627–642.
33.
ZhaoMHZhangQYPanE, et al. (2014) Fundamental solutions and numerical modeling of an elliptical crack with polarization saturation in a transversely isotropic piezoelectric medium. Engineering Fracture Mechanics131: 627–642.
34.
ZhouZGWangB (2002) Investigation of anti-plane shear behavior of two collinear cracks in the piezoelectric materials by using the non-local theory. International Journal of Solids and Structures39: 1731–1742.
35.
ZhouZGLangJWuLZ (2006a) The nonlocal theory solution of a mode-I crack in functionally graded materials subjected to harmonic stress waves. Journal of Mechanics of Materials and Structures1: 101–123.
36.
ZhouZGSunJLWangB (2004) Investigation of the behavior of a crack in a piezoelectric material subjected to a uniform tension loading by use of the non-local theory. International Journal of Engineering Science42: 2041–2063.
37.
ZhouZGWuLZDuSY (2006b) Non-local theory solution for a mode I crack in piezoelectric materials. European Journal of Mechanics A: Solids25: 793–807.
38.
ZhouZGWuLZDuSY (2007a) Non-local theory solution of two mode-I collinear cracks in the piezoelectric materials. Mechanics of Advanced Materials and Structures14: 191–201.
39.
ZhouZGZhangPWWuLZ (2007b) Two parallel limited-permeable mode-I cracks or four parallel limited-permeable mode-I cracks in the piezoelectric materials. International Journal of Solids and Structures44: 4184–4205.
