Symplectic Isogeometric Analysis Coupling Method for Interfacial Fracture of Piezoelectric Quasicrystal Composites With Notches

YANG Zhenting; WANG Yajing; NIE Xueyang; XU Xinsheng; ZHOU Zhenhuan

doi:10.21656/1000-0887.440247

Volume 45 Issue 2

Feb. 2024

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2024 > 45(2): 144-154

YANG Zhenting, WANG Yajing, NIE Xueyang, XU Xinsheng, ZHOU Zhenhuan. Symplectic Isogeometric Analysis Coupling Method for Interfacial Fracture of Piezoelectric Quasicrystal Composites With Notches[J]. Applied Mathematics and Mechanics, 2024, 45(2): 144-154. doi: 10.21656/1000-0887.440247

Citation:

PDF( 1789 KB)

Symplectic Isogeometric Analysis Coupling Method for Interfacial Fracture of Piezoelectric Quasicrystal Composites With Notches

doi: 10.21656/1000-0887.440247

State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian, Liaoning 116024, P.R.China

Received Date: 2023-08-17
Rev Recd Date: 2023-10-12
Publish Date: 2024-02-01

Abstract

Abstract

A high-precision semi numerical and semi analytical method for interfacial fracture problem of piezoelectric quasicrystals (PQCs)/piezoelectric crystals (PZCs)/elastic material composites with notches was developed. Firstly, the Hamiltonian system was introduced and the Hamiltonian dual equations for the 3-material composite were formulated. The higher order partial differential governing equations were transformed into a set of ordinary differential equations. Secondly, the symplectic eigenvalues and eigensolutions were obtained through separation of variables. The physical quantities were expressed with the expansion of symplectic series. Finally, a symplectic isogeometric analysis (IGA) coupling equation was derived through combination of the symplectic series and the IGA. The analytical expressions of the physical quantities near the notch tip and the intensity factors were derived.
- quasicrystal,
- piezoelectric material,
- isogeometric analysis,
- Hamiltonian system,
- V notch,
- interfacial fracture

FullText(HTML)

References(27)

References

[1]	DING D H, YANG W G, HU C Z, et al. Generalized elasticity theory of quasi-crystals[J]. Physical Review B, 1993, 48(10): 7003-7010. doi: 10.1103/PhysRevB.48.7003
[2]	LI X F, FAN T Y, SUN Y F. A decagonal quasicrystal with a Griffith crack[J]. Philosophical Magazine A, 1999, 79(8): 1943-1952. doi: 10.1080/01418619908210401
[3]	FAN T Y, MAI Y W. Elasticity theory, fracture mechanics, and some relevant thermal properties of quasi-crystalline materials[J]. Applied Mechanics Reviews, 2004, 57(5): 325-343. doi: 10.1115/1.1763591
[4]	GUO J H, YU J, XING Y M. Anti-plane analysis on a finite crack in a one-dimensional hexagonal quasicrystal strip[J]. Mechanics Research Communications, 2013, 52: 40-45. doi: 10.1016/j.mechrescom.2013.06.005
[5]	LI L H, LIU G T. Icosahedral quasicrystals solids with an elliptic hole under uniform heat flow[J]. Chinese Physics B, 2014, 23(5): 056101. doi: 10.1088/1674-1056/23/5/056101
[6]	苏梦雨, 肖俊华, 冯国益. 一维六方准晶中纳米尺度开裂孔洞的Ⅲ型断裂力学[J]. 固体力学学报, 2020, 41(3): 281-292. https://www.cnki.com.cn/Article/CJFDTOTAL-GTLX202003011.htm SU Mengyu, XIAO Junhua, FENG Guoyi. Type Ⅲ fracture mechanics of a nanoscale cracked hole in one-dimensional hexagonal quasicrystal[J]. Chinese Journal of Solid Mechanics, 2020, 41(3): 281-292. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-GTLX202003011.htm
[7]	范天佑. 准晶数学弹性力学和缺陷力学[J]. 力学进展, 2000, 30(2): 161-174. https://www.cnki.com.cn/Article/CJFDTOTAL-LXJZ200002000.htm FAN Tianyou. Mathematical theory of elasticity and defects of quasicrystals[J]. Advances in Mechanics, 2000, 30(2): 161-174. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-LXJZ200002000.htm
[8]	GAO Y, RICOEUR A, ZHANG L L, et al. Crack solutions and weight functions for plane problems in three-dimensional quasicrystals[J]. Archive of Applied Mechanics, 2014, 84(8): 1103-1115. doi: 10.1007/s00419-014-0868-4
[9]	WANG X, SCHIAVONE P. Elastic field for a blunt crack in a decagonal quasicrystalline material[J]. Engineering Fracture Mechanics, 2019, 220: 106657. doi: 10.1016/j.engfracmech.2019.106657
[10]	HU K, YANG W, FU J, et al. Analysis of an anti-plane crack in a one-dimensional orthorhombic quasicrystal strip[J]. Mathematics and Mechanics of Solids, 2022, 27(11): 2467-2479. doi: 10.1177/10812865211073814
[11]	LI L H, LIU G T. Decagonal quasicrystal plate with elliptic holes subjected to out-of-plane bending moments[J]. Physics Letters A, 2014, 378(10): 839-844. doi: 10.1016/j.physleta.2014.01.024
[12]	张炳彩, 丁生虎, 张来萍. 一维六方准晶双材料中圆孔边共线界面裂纹的反平面问题[J]. 应用数学和力学, 2022, 43(6): 639-647. doi: 10.21656/1000-0887.420202 ZHANG Bingcai, DING Shenghu, ZHANG Laiping. The anti-plane problem of collinear interface cracks emanating from a circular hole in 1D hexagonal quasicrystal bi-materials[J]. Applied Mathematics and Mechanics, 2022, 43(6): 639-647. (in Chinese) doi: 10.21656/1000-0887.420202
[13]	卢绍楠, 赵雪芬, 马园园. 一维六方压电准晶双材料界面共线裂纹问题[J]. 应用数学和力学, 2023, 44(7): 809-824. doi: 10.21656/1000-0887.430111 LU Shaonan, ZHAO Xuefen, MA Yuanyuan. Research on interfacial collinear cracks between 1D hexagonal piezoelectric quasicrystal bimaterials[J]. Applied Mathematics and Mechanics, 2023, 44(7): 809-824. (in Chinese) doi: 10.21656/1000-0887.430111
[14]	JIANG L J, LIU G T. The interaction between a screw dislocation and a wedge-shaped crack in one-dimensional hexagonal piezoelectric quasicrystals[J]. Chinese Physics B, 2017, 26(4): 044601. doi: 10.1088/1674-1056/26/4/044601
[15]	LI Y D, BAO R, CHEN W. Axial shear fracture of a transversely isotropic piezoelectric quasicrystal cylinder: which field (phonon or phason) has more contribution?[J]. European Journal of Mechanics A: Solids, 2018, 71: 179-186. doi: 10.1016/j.euromechsol.2018.03.019
[16]	ZHOU Y B, LI X F. Fracture analysis of an infinite 1D hexagonal piezoelectric quasicrystal plate with a penny-shaped dielectric crack[J]. European Journal of Mechanics A: Solids, 2019, 76: 224-234. doi: 10.1016/j.euromechsol.2019.04.011
[17]	ZHAO M, DANG H, FAN C, et al. Analysis of a three-dimensional arbitrarily shaped interface crack in a one-dimensional hexagonal thermo-electro-elastic quasicrystal bi-material, part 1: theoretical solution[J]. Engineering Fracture Mechanics, 2017, 179: 59-78. doi: 10.1016/j.engfracmech.2017.04.019
[18]	DANG H, ZHAO M, FAN C, et al. Analysis of a three-dimensional arbitrarily shaped interface crack in a one-dimensional hexagonal thermo-electro-elastic quasicrystal bi-material, part 2: numerical method[J]. Engineering Fracture Mechanics, 2017, 180: 268-281. doi: 10.1016/j.engfracmech.2017.05.042
[19]	HU K, GAO C, ZHONG Z, et al. Interaction of collinear interface cracks between dissimilar one-dimensional hexagonal piezoelectric quasicrystals[J]. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 2021, 101(11): e202000360. doi: 10.1002/zamm.202000360
[20]	HUGHES T J R, COTTRELL J A, BAZILEVS Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement[J]. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39): 4135-4195.
[21]	姚伟岸, 钟万勰. 辛弹性力学[M]. 北京: 高等教育出版社, 2002. YAO Weian, ZHONG Wanxie. Symplectic Elasticity[M]. Beijing: Higher Education Press, 2002. (in Chinese)
[22]	XU C H, ZHOU Z H, XU X S, et al. Electroelastic singularities and intensity factors for an interface crack in piezoelectric-elastic bimaterials[J]. Applied Mathematical Modelling, 2015, 39(9): 2721-2739. doi: 10.1016/j.apm.2014.10.061
[23]	ZHOU Z, YANG Z, XU W, et al. Evaluation of electroelastic singularity of finite-size V-notched one-dimensional hexagonal quasicrystalline bimaterials with piezoelectric effect[J]. Theoretical and Applied Fracture Mechanics, 2019, 100: 139-153.
[24]	WU Y F, CHEN W Q, LI X Y. Indentation on one-dimensional hexagonal quasicrystals: general theory and complete exact solutions[J]. Philosophical Magazine, 2013, 93(8): 858-882. doi: 10.1080/14786435.2012.735772
[25]	LI X Y, LI P D, WU T H, et al. Three-dimensional fundamental solutions for one-dimensional hexagonal quasicrystal with piezoelectric effect[J]. Physics Letters A, 2014, 378(10): 826-834. doi: 10.1016/j.physleta.2014.01.016
[26]	YANG J, LI X. Analytic solutions of problem about a circular hole with a straight crack in one-dimensional hexagonal quasicrystals with piezoelectric effects[J]. Theoretical and Applied Fracture Mechanics, 2016, 82: 17-24.
[27]	CHEN C D, CHUE C H. Singular electro-mechanical fields near the apex of a piezoelectric bonded wedge under antiplane shear[J]. International Journal of Solids and Structures, 2003, 40(23): 6513-6526.