ORIGINAL CONTRIBUTION
On higher order parameters in cracked composite plates under far-field pure shear
Saeid Ghouli,
Majid R. Ayatollahi,
Morteza Nejati,
Saeid Ghouli
Fatigue and Fracture Research Laboratory, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran
Search for more papers by this authorMajid R. Ayatollahi
Fatigue and Fracture Research Laboratory, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran
Search for more papers by this authorCorresponding Author
Morteza Nejati
Department of Earth Sciences, ETH Zurich, Zurich, Switzerland
Correspondence: Morteza Nejati, Department of Earth Sciences, ETH Zurich, Zurich, Switzerland.
Email: [email protected]
Search for more papers by this authorSaeid Ghouli,
Majid R. Ayatollahi,
Morteza Nejati,
Saeid Ghouli
Fatigue and Fracture Research Laboratory, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran
Search for more papers by this authorMajid R. Ayatollahi
Fatigue and Fracture Research Laboratory, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran
Search for more papers by this authorCorresponding Author
Morteza Nejati
Department of Earth Sciences, ETH Zurich, Zurich, Switzerland
Correspondence: Morteza Nejati, Department of Earth Sciences, ETH Zurich, Zurich, Switzerland.
Email: [email protected]
Search for more papers by this authorAbstract
This paper presents the analytical solution of the crack tip fields as well as the crack parameters in an infinitely large composite plate with a central crack subjected to pure shear loading. To this end, the complex variable method is employed to formulate an asymptotic solution for the crack tip fields in an anisotropic plane. Using a stress-based definition of the crack tip modes of loading, only the mode II crack parameters are found to be non-zero under pure shear load. Special focus is given to the determination of the higher order parameters of the crack tip asymptotic field, particularly the first non-singular term, ie, the T-stress. Unlike the isotropic materials, in which the T-stress is zero under pure shear, it is found that the T-stress is non-zero for the case of anisotropic materials, being the only material-dependent crack tip stress parameter. The veracity of our exact crack tip fields is assessed and verified through a comparison made with respect to the finite element (FE) solution. Finally, we demonstrate the significance of the T-stress on stresses near the crack tip in composite plates under pure shear loads.
References
- 1Sih GC, Paris PC, Irwin GR. On cracks in rectilinearly anisotropic bodies. Int J Fract Mech. 1965; 1(3): 189–203.
- 2Ayari ML, Ye Z. Maximum strain theory for mixed mode crack propagation in anisotropic solids. Eng Fract Mech. 1995; 52(3): 389–400.
- 3Azhdari A, Nemat-Nasser S. Energy-release rate and crack kinking in anisotropic brittle solids. J Mech Phys Sol. 1996; 44(6): 929–951.
- 4Gao H, Chiu CH. Slightly curved or kinked cracks in anisotropic elastic solids. Int J Sol Struct. 1992; 29(8): 947–972.
- 5Saouma VE, Ayari ML, Leavell DA. Mixed mode crack propagation in homogeneous anisotropic solids. Eng Fract Mech. 1987; 27(2): 171–184.
- 6Ye Z, Ayari M. Prediction of crack propagation in anisotropic solids. Eng Fract Mech. 1994; 49(6): 797–808.
- 7Cotterell B, Rice J. Slightly curved or kinked cracks. Int J Fract. 1980; 16(2): 155–169.
- 8Smith DJ, Ayatollahi MR, Pavier MJ. The role of T-stress in brittle fracture for linear elastic materials under mixed-mode loading. Fatigue Fract Eng Mat Struct. 2001; 24(2): 137–150.
- 9Azhdari A, Obata M, Nemat-Nasser S. Alternative solution methods for crack problems in plane anisotropic elasticity, with examples. Int J Sol Struct. 2000; 37(44): 6433–6478.
- 10Bao G, Ho S, Suo Z, Fan B. The role of material orthotropy in fracture specimens for composites. Int J Sol Struct. 1992; 29(9): 1105–1116.
- 11Barnett D, Asaro R. The fracture mechanics of slit-like cracks in anisotropic elastic media. J Mech Phys Sol. 1972; 20(6): 353–366.
- 12Beom H, Cui C, Jang H. Application of an orthotropy rescaling method to edge cracks and kinked cracks in orthotropic two-dimensional solids. Int J Eng Sci. 2012; 51: 14–31.
- 13Bowie OL, Freese CE. Central crack in plane orthotropic rectangular sheet. Int J Fract Mech. 1972; 8(1): 49–57.
- 14Chalivendra VB. Mode-I crack-tip stress fields for inhomogeneous orthotropic medium. Mech Mat. 2008; 40(4-5): 293–301.
- 15Chaudhuri RA. Three-dimensional singular stress field at the front of a crack and lattice crack deviation (LCD) in a cubic single crystal plate. Philos Mag. 2010; 90(15): 2049–2113.
- 16Cheng ZQ, Reddy J. Laminated anisotropic thin plate with an elliptic inhomogeneity. 647–657. 2004; 36(7).
- 17Delale F, Erdogan F. The problem of internal and edge cracks in an orthotropic strip. J Appl Mech. 1977; 44(2): 237–242.
- 18Guo L-C, Wu L-Z, Zeng T, Ma L. Mode I crack problem for a functionally graded orthotropic strip. European J Mech Sol. 2004; 23(2): 219–234.
- 19Hajimohamadi M, Ghajar R. Stress intensity factors for cracks emanating from a circular hole in an infinite quasi-orthotropic plane. Fatigue Fract Eng Mat Struct. 2019; 42(3): 743–751.
- 20Hassani A, Monfared M. Analysis of an orthotropic circular bar weakened by multiple radial cracks under torsional transient loading. Eng Fract Mech. 2017; 186: 300–315.
- 21Hoenig A. Near-tip behavior of a crack in a plane anisotropic elastic body. Eng Fract Mech; 16(3): 393–403.
- 22Hwu C. Collinear cracks in anisotropic bodies. Int J Fract. 1991; 52(4): 239–256.
- 23Ju S, Chiu C, Jhao B. Determination of SIFs, crack-tip coordinates and crack angle of anisotropic materials. Fatigue Fract Eng Mat Struct. 2009; 33(1): 43–53.
- 24Kaya AC, Erdogan F. Stress intensity factors and COD in an orthotropic strip. Int J Fract. 1980; 16(2): 171–190.
- 25Li X-F, Lee K. Closed-form solution for an orthotropic elastic strip with a crack perpendicular to the edges under arbitrary anti-plane shear. ZAMM. 2009; 89(5): 370–382.
- 26Monfared M, Bagheri R, Yaghoubi R. The mixed mode analysis of arbitrary configuration of cracks in an orthotropic FGM strip using the distributed edge dislocations. Int J Sol Struct. 2018; 130-131: 21–35.
- 27Nobile L, Piva A, Viola E. On the inclined crack problem in an orthotropic medium under biaxial loading. Eng Fract Mech. 2004; 71(4-6): 529–546.
- 28Obata M, Nemat-Nasser S, Goto Y. Branched cracks in anisotropic elastic solids. J Appl Mech. 1989; 56(4): 858–864.
- 29Ozturk M, Erdogan F. Mode I crack problem in an inhomogeneous orthotropic medium. Int J Eng Sci. 1997; 35(9): 869–883.
- 30Rabbolini S, Pataky GJ, Sehitoglu H, Beretta S. Fatigue crack growth in Haynes 230 single crystals: an analysis with digital image correlation. Fatigue Fract Eng Mat Struct. 2015; 38(5): 583–596.
- 31Sharma DS, Dave JM. Stress intensity factors for hypocycloidal hole with cusps in infinite anisotropic plate. Theo Appl Fract Mech. 2015; 75: 44–52.
- 32Suo Z. Singularities, interfaces and cracks in dissimilar anisotropic media. Proc Royal Soc Math Phys Eng Sci. 1990; 427(1873): 331–358.
- 33Tafreshi A. A unified and integrated numerical-analytical approach for evaluation of Jk-integrals of an interfacial crack in orthotropic and isotropic bimaterials. Fatigue Fract Eng Mat Struct. 2018; 41(9): 1963–1979.
- 34Tian WY, Rajapakse RKND. Fracture analysis of magnetoelectroelastic solids by using path independent integrals. Int J Fract. 2005; 131(4): 311–335.
- 35Willis J. Fracture mechanics of interfacial cracks. J Mech Phys Sol. 1971; 19(6): 353–368.
- 36Wu B-H, Erdogan F. The surface and through crack problems in orthotropic plates. Int J Sol Struct. 1989; 25(2): 167–188.
- 37Wu EM. Application of fracture mechanics to anisotropic plates. J Appl Mech. 1967; 34(4): 967–974.
- 38Wu K-C. Representations of stress intensity factors by path-independent integrals. J Appl Mech. 1989; 56(4): 780–785.
- 39Wu KC. Stress intensity factors and energy release rate for anisotropic plates based on the classical plate theory. Compos Part Eng. 2016; 97: 300–308.
- 40Kim J, Moon H, Earmme Y. Inplane and antiplane T-stresses for an interface crack in anisotropic bimaterial. Mech Mat. 2001; 33(1): 21–32.
- 41Ma H, Zhao L, Chen Y. Non-singular terms for multiple cracks in anisotropic elastic solids. Theo Appl Fract Mech. 1997; 27(2): 129–134.
- 42Rungamornrat J, Pinitpanich M. T-stress solution of penny-shaped cracks in transversely isotropic elastic media. Eng Fract Mech. 2016; 158: 194–208.
- 43Yang S, Yuan FG. Determination and representation of the stress coefficient terms by path-independent integrals in anisotropic cracked solids. Int J Fract. 2000; 101(4): 291–319.
- 44Zhang L, Kim J-H. A complex variable approach for asymptotic mode-III crack-tip fields in an anisotropic functionally graded material. Eng Fract Mech. 2009; 76(16): 2512–2525.
- 45Lee K-H, Hawong J-S, Choi S-H. Dynamic stress intensity factors KI, KII and dynamic crack propagation characteristics of orthotropic material. Eng Fract Mech. 1996; 53(1): 119–140.
- 46Yang S, Yuan F-G. Kinked crack in anisotropic bodies. Int J Sol Struct. 2000; 37(45): 6635–6682.
- 47Brandinelli L, Massabò R. Mode II weight functions for isotropic and orthotropic double cantilever beams. Int J Fract. 2006; 139(1): 1–25.
- 48Craciun E, Sadowski T, Rbâea A. Stress concentration in an anisotropic body with three equal collinear cracks in mode II of fracture. I. Analytical study. ZAMM—Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik. 2014; 94(9): 721–729.
- 49Cui D, Zhang X. The solution of stress intensity factor for mode II cracking in orthotropic cracked plate by complex variable-variational method. Comput Mech. 1991; 7(5-6): 361–368.
10.1007/BF00350165 Google Scholar
- 50Alizadeh F, Guedes Soares C. Experimental and numerical investigation of the fracture toughness of Glass/Vinylester composite laminates. European J Mech Sol. 2019; 73(August 2018): 204–211.
- 51Barrett J, Foschi R. Mode II stress-intensity factors for cracked wood beams. Eng Fract Mech. 1977; 9(2): 371–378.
- 52Chow W, Atluri S. Stress intensity factors as the fracture parameters for delamination crack growth in composite laminates. Compos Part B Eng. 1997; 28(4): 375–384.
- 53Kenane M, Benmedakhene S, Azari Z. Fracture and fatigue study of unidirectional glass/epoxy laminate under different mode of loading. Fatigue Fract Eng Mat Struct. 2010; 33(5): 284–293.
- 54Kim H-B, Naito K, Oguma H. Mode II fracture toughness of two-part acrylic-based adhesive in an adhesively bonded joint: end-notched flexure tests under static loading. Fatigue Fract Eng Mat Struct. 2017; 40(11): 1795–1808.
- 55Lakshminarayana H. A symmetric rail shear test for mode II fracture toughness (GIIC) of composite materials–finite element analysis. J Compos Mat. 1984; 18(3): 227–238.
- 56Liu C. Analytical stress around mode II crack parallel to principle axis in orthotropic composite plate. Eng Fract Mech. 1996; 6: 791–803.
- 57Nwosu S, Hui D, Dutta P. Dynamic mode II delamination fracture of unidirectional graphite/epoxy composites. Compos Part Eng. 2003; 34(3): 303–316.
- 58Thangjitham S, Choi H. Interlaminar crack problems of a laminated anisotropic medium. Int J Sol Struct. 1993; 30(7): 963–980.
- 59Yao XF, Zhao HP, Yeh HY. Dynamic caustic analysis of propagating mode II cracks in transversely isotropic material. J Reinf Plast Compos. 2005; 24(6): 657–667.
- 60Lekhnitskii SG. Anisotropic Plates. New York, USA:Gordon and Breach Science Publishers; 1968.
- 61Saint Venant B. Sur la distributiondes élasticités autour de chaque point d'un solide ou d'unmilieu de contexture quelconque, particulièrement lorsqu'il est amorphe sans être isotrope. J de Math Pures et Appliquées. 1863; VIII(2): 257–430.
- 62Worotnicki G. Comprehensive rock engineering. Pergamon, Oxford, Ch. CSIRO Triaxial Stress Measure Cell. 1993; 3: 329–394.
- 63Aminzadeh A, Fahimifar A, Nejati M. On Brazilian disk test for mixed-mode I/II fracture toughness experiments of anisotropic rocks. Theo Appl Fract Mech. 2019; 102(April): 222–238.
- 64Nejati M, Aminzadeh A, Saar MO, Driesner T. Modified semi-circular bend test to determine the fracture toughness of anisotropic rocks. Eng Fract Mech. 2019; 213(February): 153–171.
- 65Theocaris PS, Spyropoulos CP. Photoelastic determination of complex stress intensity factors for slant cracks under biaxial loading with higher-order term effects. Acta Mechanica. 1983; 48(1-2): 57–70.
- 66Ayatollahi MR, Nejati M. An over-deterministic method for calculation of coefficients of crack tip asymptotic field from finite element analysis. Fatigue Fract Eng Mat Struct. 2011; 34(3): 159–176.
- 67Liu N, Jeffers AE. A geometrically exact isogeometric Kirchhoff plate: feature-preserving automatic meshing and C1 rational triangular Bézier spline discretizations. Int J Num Method Eng. 2018; 115(3): 395–409.
- 68Liu N, Jeffers AE. Feature-preserving rational Bézier triangles for isogeometric analysis of higher-order gradient damage models. Comput Method Appl Mech Eng. 2019; 357: 112585.
- 69Tsai S, Hahn H. Introduction to Composite Materials. Lancaster:Technomic Publishing Company, Inc; 1980.
- 70Bažant Z., Estenssoro L. Surface singularity and crack propagation. Int J Sol Struct. 1979; 15: 405–426.
- 71Benthem J. State of stress at the vertex of a quarter-infinite crack in a half-space. Int J Sol Struct. 1977; 13(5): 479–492.
- 72Nakamura T, Parks D. Three-dimensional stress field near the crack front of a thin elastic plate. J Appl Mech. 1988; 55: 805–813.
- 73Nakamura T, Parks D. Antisymmetrical 3-D stress field near the crack front of a thin elastic plate. Int J Sol Struct. 1989; 25(12): 1411–1426.
- 74Kotousov a., Lazzarin P, Berto F, Pook L. Three-dimensional stress states at crack tip induced by shear and anti-plane loading. Eng Fract Mech. 2013; 108: 65–74.
- 75Nejati M, Paluszny A, Zimmerman RW. A disk-shaped domain integral method for the computation of stress intensity factors using tetrahedral meshes. Int J Sol Struct. 2015; 69-70: 230–251.
- 76Pook L, Berto F, Campagnolo A, Lazzarin P. Coupled fracture mode of a cracked disc under anti-plane loading. Eng Fract Mech. 2014; 128: 22–36.
- 77Erdogan F, Sih GC. On the crack extension in plates under plane loading and transverse shear. J Basic Eng. 1963; 85(4): 519.
10.1115/1.3656897 Google Scholar
- 78Lim W-K, Choi S-Y, Sankar BV. Biaxial load effects on crack extension in anisotropic solids. Eng Fract Mech. 2001; 68(4): 403–416.
- 79Romanowicz M. A non-local stress fracture criterion accounting for the anisotropy of the fracture toughness. Eng Fract Mech. 2019; 214(April): 544–557.
- 80Cahill L, Natarajan S, Bordas S, O'Higgins R, McCarthy C. An experimental/numerical investigation into the main driving force for crack propagation in uni-directional fibre-reinforced composite laminae. Compos Struct. 2014; 107(1): 119–130.
- 81Koohbor B, Mallon S, Kidane A, Sutton MA. A DIC-based study of in-plane mechanical response and fracture of orthotropic carbon fiber reinforced composite. Compos Part Eng. 2014; 66: 388–399.
