**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30121

##### Non-Local Behavior of a Mixed-Mode Crack in a Functionally Graded Piezoelectric Medium

**Authors:**
Nidhal Jamia,
Sami El-Borgi

**Abstract:**

In this paper, the problem of a mixed-Mode crack embedded in an infinite medium made of a functionally graded piezoelectric material (FGPM) with crack surfaces subjected to electro-mechanical loadings is investigated. Eringen’s non-local theory of elasticity is adopted to formulate the governing electro-elastic equations. The properties of the piezoelectric material are assumed to vary exponentially along a perpendicular plane to the crack. Using Fourier transform, three integral equations are obtained in which the unknown variables are the jumps of mechanical displacements and electric potentials across the crack surfaces. To solve the integral equations, the unknowns are directly expanded as a series of Jacobi polynomials, and the resulting equations solved using the Schmidt method. In contrast to the classical solutions based on the local theory, it is found that no mechanical stress and electric displacement singularities are present at the crack tips when nonlocal theory is employed to investigate the problem. A direct benefit is the ability to use the calculated maximum stress as a fracture criterion. The primary objective of this study is to investigate the effects of crack length, material gradient parameter describing FGPMs, and lattice parameter on the mechanical stress and electric displacement field near crack tips.

**Keywords:**
Functionally graded piezoelectric material,
mixed-mode crack,
non-local theory,
Schmidt method.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1126844

**References:**

[1] Gautschi, G., 2002. Piezoelectric Sensorics. Springer-Verlag Berlin Heidelberg.

[2] Deeg, W., 1980. The analysis of dislocation, crack and inclusion problems in piezoelectric solids. Ph.D. thesis, Stanford University.

[3] Sosa, H., 1991. Plane Problems in Piezoelectric Media with Defects. International Journal of Solids and Structures 28, 491-505.

[4] F. Erdogan, 1995. Fracture mechanics of functionally graded materials, Compos. Eng.5 753–770.

[5] Chue, C.H., Ou, Y.L., 2005. Mode III crack problems for two bonded functionally graded piezoelectric materials, International Journal of Solids and Structures 42, 3321–3337.

[6] Hsu, W.H., Chue, C.H., 2008. Mode III fracture problem of an arbitrarily oriented crack in a FGPM strip bonded to a FGPM half plane. International Journal of Solids and Structures 45, 6333–6346.

[7] Chen, Y.J., Chue, C.H., 2010. Mode III fracture problem of a cracked FGPM surface layer bonded to a cracked FGPM substrate. Archive of Applied Mechanics 80. Issue 3. 285-305.

[8] Ueda, S., 2007. Electromechanical impact of an impermeable parallel crack in a functionally graded piezoelectric strip. European Journal of Mechanics - A/Solids 26. 123-136.

[9] Zhou, Z. G., Chen, Z. T., 2008. The interaction of two parallel Mode-I limited-permeable cracks in a functionally graded piezoelectric material. European Journal of Mechanics A/Solid. 27. 824–846.

[10] Eringen, A.C., 1972. Nonlocal polar elastic continua. International Journal of Engineering Science 10, 1-16.

[11] Eringen, A.C., Speziale, C.G., Kim, B.S., 1977. Crack tip problem in nonlocal elasticity, Journal of the Mechanics and Physics of Solids 25, 339–355.

[12] El-Borgi, S., Keer, L. Ben Said, W., 2004. An embedded crack in a functionally graded coating bonded to a homogeneous substrate under frictional Hertzian contact, Wear 257, 760–776.

[13] Zhou, Z.G., Sun, J.L., Wang, B., 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 Science, 42 (19-20). 2041–63.

[14] Jamia N., El-Borgi S., Rekik M., Usman S., 2014. Investigation of the behavior of a mixed-Mode crack in a functionally graded magneto–electro-elastic material by use of the non-local theory. Theoretical and Applied Fracture Mechanics 74, 126–142.

[15] Rekik, M., El-Borgi, S., Ounaies, Z., 2014. An Axisymmetric Problem of an Embedded Mixed-Mode Crack in a Functionally Graded Magnetoelectroelastic Infinite Medium". Journal of Applied Mathematical Modelling, 38, 1193–1210.

[16] Erdogan, F., Gupta, G.D. and Cook, T.S., 1973. Numerical solution of singular integral equations, 368-425, in Mechanics of Fracture, edited by G. Sih, Norrdhoff, Leyden.

[17] Morse, P.M., Feshbach, H., 1985. Methods of Theoretical Physics, McGraw-Hill, New York, pp. 926–1010.

[18] Yan, W.F., 1967. Axisymmetric slipless indentation of an infinite elastic cylinder, SIAM Journal of Applied Mathematics 15, 219–227.

[19] Gradshteyn, I.S., Ryzhik, I.M., 1980. Table of Integral, Series and Products. Academic Press, New York, pp. 1159–1161.

[20] Erdelyi, A. 1954. Tables of Integral Transforms, Vol. I. McGraw-Hill, New York.

[21] Zhou, Z.G., Zhang, P.W., Wu, L.Z., 2007. Investigation of the behavior of a Mode-I crack in functionally graded materials by use of the non-local theory, International Journal of Engineering Science 45, 242–257.

[22] Liang, J., 2009. The nonlocal theory solution of a Mode-I crack in functionally graded materials. Science in China Series E: Technological Sciences 52 (4).

[23] Eringen A.C., 1983. Interaction of a dislocation with a crack. Journal of Applied Physics 54(12), 6811–6817.

[24] Kim, B.S., Eringen, A.C., 1973. Stress distribution around an elliptic hole in an infinite micropolar elastic plate. Letter in Applied and Engineering Sciences. 1, 381-390.

[25] Griffith, A.A., 1921. The phenomenon of rupture and flow in solids. Philosophical Transactions of the Royal Society. A221, 163.

[26] Eringen, A.C., 1978. Linear crack subject to shear, International Journal of Fracture 14, 367–379.

[27] Kuna, M., 2010. Fracture mechanics of piezoelectric materials – Where are we right now?. Engineering Fracture Mechanics 77, 309-326.