Abstract
This paper presents a multi-region Trefftz boundary element method for fracture analysis in plane piezoelectricity. To model the sub-region that contains the crack, a special set of Trefftz functions that satisfy the traction-free and charge-free conditions along the crack faces are constructed. To model the remaining sub-regions, the basic set of Trefftz functions co-derived previously by the authors are employed. With the two sets of Trefftz functions, the multi-region Trefftz boundary element method is formulated by point collocation. The special set of Trefftz functions exempts all the boundary treatment of the crack faces and enables the direct determination of the electromechanical intensity factors. Numerical examples are presented to illustrate the efficacy of the formulation.
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References
Chen MC, Sze KY, Wang HT (2001) Analysis of singular stresses in bonded bimaterial wedges by computed eigen solutions and hybrid element method. Commun Numer. Meth Engrg 17:495–507
Davi G, Milazzo A (2001) Multidomain boundary integral formulation for piezoelectric materials fracture mechanics. Inter J Solids and Struct 38:7065–7078
Ding HJ, Wang GQ, Chen WQ (1998) A boundary integral formulation and 2D fundamental solutions for piezoelectric media. Comp Meth Appl Mech & Engrg 158:65–80
Domingues JS, Portela A, Castro PMST (1999) Trefftz boundary element method applied to fracture mechanics. Engrg Fracture Mech 64:67–86
Eubanks RA, Sternberg E (1954) On the axisymmetric problem of elastic theory for a medium with transverse isotropy. J Rational Mech Anal 3:89–101
Freitas JAT, Ji ZY (1996) Hybrid-Trefftz finite element formulation for simulation of singular stress fields. Inter J Numer Meth Engrg 39:281–308
Herrera I (1984) Trefftz method. In: Brebbia CA (ed), Topics in Boundary Element Research–Basic Principles and Applications, Springer, Berlin
Jin WG, Cheung YK, Zienkiewicz OC (1990) Application of the Trefftz method in plane elasticity problems. Inter J Numer Meth Engrg 30:1147–1161
Jin WG, Sheng N, Sze KY, Li J (2005) Trefftz indirect methods for plane piezoelectricity. Inter J Numer Meth Engng 63:139–158
Jirousek J, Wroblewski A (1996) T-elements: state of the art and future trends. Arch Comput Meth Engrg 3–4:323–434
Kita E, Kamiya N (1995) Trefftz method: an overview. Adv Engrg Software 24:3–12
Kompis V (ed.) (2003) Selected Topics in Boundary Integral Formulations for Solids and Fluids. Springer, NY
Leitao VMA, (1998) Applications of multi-region Trefftz-collocation to fracture mechanics. Engrg Anal Boundary Elem 22:251–256
Pak YE (1992) Linear electro-elastic fracture mechanics of piezoelectric materials. Inter J Fract 54:79–100
Pan E (1999) A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric solids. Engrg Anal Boundary Elem 23:67–76
Park SB, Sun CT (1995) Effect of electric field on fracture of piezoelectric ceramics. Inter J Fracture 70:203–216
Piltner R, (1985) Special finite elements with holes and internal cracks. Inter J Numer Meth Engrg 21:1471–1485
Portela A, Charafi A (1997a) Programming Trefftz boundary elements. Advances in Engrg Software 28:509–523
Portela A, Charafi A (1997b) Trefftz boundary element method for domains with slits. Engrg Anal Boundary Elem 20:299–304
Portela A, Charafi A (1999) Trefftz boundary elements–multi-region formulations. Inter J Numer Meth Engrg 45:821–840
Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical Recipes in Fortran 77, 2nd edn. Cambridge University Press
Qin QH, (2003a) Variational formulations for TFEM of piezoelectricity. Inter J Solids and Struct 40:6335–6346
Qin QH, (2003b) Solving anti-plane problems of piezoelectric materials by the Trefftz finite element approach. Comput Mech 31:461–468
Rajapakse RKND, Xu XL (2001) Boundary element modeling of cracks in piezoelectric solids. Engrg Anal Boundary Elem 25:771–781
Shindo Y, Watanabe K, Narita F (2000) Electroelastic analysis of a piezoelectric ceramic strip with a central crack. Inter J Engrg Science 38:1–19
Sosa HA, (1992) On the fracture mechanics of piezoelectric solids. Inter J Solids and Struct 29:2613–2622
Sze KY, Jin WG, Sheng N, Li J (2003) Trefftz methods for plane piezoelectricity. Comput Assist Mech & Engrg Sciences 10:619–627
Wang BL, Mai YW (2002) A piezoelectric material strip with a crack perpendicular to its boundary surfaces. Inter J Solids and Struct 39:4501–4524
Zhang HQ, (1978) A united theory on general solutions of systems of elasticity equations. J Dalian Univ of Tech 3:23–47 (in Chinese)
Zielinski AP, Herrera I (1987) Trefftz method: fitting boundary conditions. Inter J Numer Meth Engrg 24:871–891
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Sheng, N., Sze, K.Y. Multi-region Trefftz boundary element method for fracture analysis in plane piezoelectricity. Comput Mech 37, 381–393 (2006). https://doi.org/10.1007/s00466-004-0653-2
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DOI: https://doi.org/10.1007/s00466-004-0653-2