Comparative analysis of stress state in the vicinity of the crack-tips and notches using truncated Williams series expansion

R. M. Zhabbarov   ; L. V. Stepanova

Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 27 (2021) no. 4, pp. 30-67

Résumé

In the present study the asymptotic analysis of the multi-parameter Williams series expansion of the stress-strain state in the vicinity of seven different configurations is performed. The aim of the asymptotic analysis is to educate the sensitivity of different cracked and notched configurations to the number of kept terms in Williams series expansion. It is shown that all configurations with the concentrated forces possess the more sensitivity to the number of terms in the truncated Williams series expansions and, therefore, require the more terms in the series expansion. The cracked configurations with distributed loadings are less sensitive to the number of terms in Williams series expansion. As a whole, it is demonstrated that high order terms have principal role in the description of the stress-strain in the vicinity of the crack tip.

Détail

Keywords: multi-parameter Williams series expansion, Williams series coefficients high order terms, theory of functions of a complex variable.

Bibliographie

[1] Sadd M.H., Elasticity. Theory, Applications and Numerics, Academic Press, Oxford, 2021, 624 pp.

[2] Muskhelishvili N.I., Some basic problems of the mathematical theory of elasticity, Izdatel'stvo akademii nauk SSSR, M., 1954, 647 pp. (In Russ.)

[3] Shtaerman I.Ya., Contact problem in elasticity theory, Gostekhizdat, M.-L., 1949, 272 pp.

[4] Razumovsky I.A., Interference-optical methods of solid mechanics, Foundations of Engineering Mechanics, Springer, Berlin, 2011, 180 pp.

[5] Sironi R.S., Optical methods of measurement. Wholefield Techniques, CRC Press, Boca Raton, 2009, 316 pp. | DOI

[6] Ramesh K., Developments in Photoelasticity: A renaissance, IOP Publishing, 2021, 225 pp. https://iopscience.iop.org/book/mono/978-0-7503-2472-4

[7] Patil P., Vyasarayani C.P., Ramji M., “Linear least squares approach for evaluating crack tip fracture parameters using isochromatic and isoclinic data from digital photoelasticity”, Optics and Lasers in Engineering, 93 (2017), 182–194 | DOI

[8] Vivekanandan A., Ramesh K., “Study of interaction effects of asymentric cracks under biaxial loading using digital photoelasticity”, Theoretical and Applied Fracture Mechanics, 99 (2019), 104–117 | DOI

[9] Dondeti S., Tippur H.V., “A Comparative Study of Crack Branching in Glass Using Photoelasticity, Digital Image Correlation and Digital Gradient Sensing Techniques”, Challenges in Mechanics of Time Dependent Materials, Fracture, Fatigue, Failure and Damage Evolution, v. 2, Conference Proceedings of the Society for Experimental Mechanics Series, eds. Silberstein M., Amirkhizi A., Shuman X., Beese A., Berke R., Pataky G., Springer International Publishing, Cham, 2020, 9–15 | DOI

[10] Lee K.H., Kim B.S., Kim J.S., “Fabrication of transparent homogeneous functionally graded materials and crack analysis by photoelasticity”, Journal of Mechanical Science and Technology, 35 (2021), 3919–3929 | DOI

[11] Vivekanandan A., Ramesh K., “Study of Crack Interaction Effects Under Thermal Loading by Digital Photoelasticity and Finite Elements”, Experimental Mechanics, 60 (2020), 295–316 | DOI

[12] Thomre M., Ramesh K., “Evaluation of Fracture Parameters of Cracks in Compressor Blade Root Using Digital Photoelasticity”, Lecture Notes in Mechanical Engineering, 2020, 557–566 | DOI

[13] Berrekheroukh N., Sereir Z., Vivet A., Adda Bedia E.A., Fekrar A., “Experimental and numerical models to study the creep behavior of the unidirectional Alfa fiber composite strength by the photoelasticity method”, Mechanics of Time-Dependent Materials, 2021, 1573–2738 | DOI

[14] (Stevanovic) Hedrih K.R., Brcic S.V., Paunovic S., “Application of Photoelasticity to Some Nonlinear Dynamic Problems and Stress State Analysis in Dams: A Brief Overview Inspired by the Results of Prof. Vlatko Brcic”, Nonlinear Dynamics of Structures, Systems and Devices, eds. Lacarbonara W., Balachandran B., Ma J., Tenreiro Machado J., Stepan G., Springer, Cham, 2020, 357–365 | DOI

[15] Juan C. Brinez-de L., Rico-Garcia M., Restrepo-Martinez A., “PhotoelastNet: a deep convolutional neural network for evaluating the stress field by using a single color photoelasticity image”, Applied Optics, 61:7 (2022), D50—D62 | DOI

[16] Mose B., Shin D., Nam J., “Experimental Stress Analysis of Spherical Roller Bearing for High-Speed Trains Using Photoelasticity”, Experimental Techniques, 2022 | DOI

[17] Matvienko Yu.G., Two-parameter damage mechanics, Izdatel'skaya firma «Fiziko-matematicheskaya literatura», M., 2020, 210 pp. (In Russ.)

[18] Kosygina L.N., “Asymptotic representation of the stress field near the crack tip of an infinite plate with two semi-infinite symmetrical edge notches: theoretical study and computational experiment”, Vestnik Samarskogo universiteta. Estestvennonauchnaia seriia = Vestnik of Samara University. Natural Science Series, 24:2 (2018), 55–66 (In Russ.) | DOI

[19] Astafyev V.I., Radaev Yu.N., Stepanova L.V., Applied problems of fracture mechanics, Izdatel'stvo «Samarskii universitet», Samara, 1999, 195 pp. (In Russ.)

[20] Li D., Huang P., Chen Z., Yao G., Guo X., Zheng X., Yang Y., “Experimental study on fracture and fatigue crack propagation processes in concrete based on DIC technology”, Engineering Fracture Mechanics, 235 (2020), 107166 | DOI

[21] Stepanova L.V., “Experimantal determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part I”, PNRPU MECHANICS BULLETIN, 2020, no. 4, 237–249 (In Russ.) | DOI

[22] Stepanova L.V., “Experimantal determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part II”, PNRPU MECHANICS BULLETIN, 2021, no. 1, 72–85 (In Russ.) | DOI

[23] Stepanova L.V., Dolgikh V.S., “Digital processing of the results of optoelectronic measurements. The photoelasticity method and its application for determination of coefficients of the multiparameter asymptotic Williams expansion of the stress field”, Journal of Samara State Technical University. Ser. Physical and Mathematical Sciences, 21:4 (2017), 717–735 (In Russ.) | DOI

[24] Stepanova L.V., Dolgikh V.S., “Experimental determination of coefficients of a multiparameter decomposition of field of crack tip stresses: photoelasticity method”, Vestnik Samarskogo universiteta. Estestvennonauchnaia seriia = Vestnik of Samara University. Natural Science Series, 23:1 (2017), 59–68 (In Russ.)

[25] Stepanova L.V., “Photoelastic study of a double edge notched plate for determination of the Williams series expansion”, Vestnik of Samara University. Natural Sciences, 26:4 (2020), 56–67 | DOI

[26] Li Y., Zheng K., “Crack tip asymptotic field coefficients analyses based on extended finite element method using over-deterministic displacement field fitting method”, Theoretical and Applied Fracture Mecahncis, 113 (2021), 102971 | DOI

[27] Li H., Zhong H., “Weak form quadrature element analysis of crack-tip asymptotic field coefficients”, Theoretical and Applied Fracture Mechanics, 119 (2022), 103320 | DOI

[28] Williams M.L., “On the Stress Distribution at the Base of a Stationary Crack”, Journal of Applied Mechanics, 24:1 (1956), 109–114 http://authors.library.caltech.edu/47558/1/382747.pdf

[29] Williams M.L., “Stress Singularities Resulting From Various Boundary Conditions in Angular Corners of Plates in Extension”, Journal of Applied Mechanics, 74:4 (1952), 526–528 http://authors.library.caltech.edu/47672/1/382785.pdf

[30] Karihaloo B.L., Abdalla H., Xiao Q.Z., “Coefficients of the crack tip asymptotic field for wedge splitting specimens”, Engineering Fracture Mechanics, 70:17 (2003), 2407–2420 | DOI

[31] Karihaloo B.L., Xiao Q.Z., “Accurate determination of the coefficients of elastic crack tip asymptotic field by a hybrid crack element with p-adaptivity”, Engineering Fracture Mechanics, 68:15 (2001), 1609–1630 | DOI

[32] Karihaloo B.L., Xiao Q.Z., “Higher order terms of the crack tip asymptotic field for a wedge-splitting specimen”, International Journal of Fracture, 112 (2001), 129–137 | DOI

[33] Hello G., “Analytical determination of coefficients in crack-tip stress expansions for a finite crack in an infinite plane medium.”, International Journal of Solids and Structures, 49 (2012), 556–566 | DOI

[34] Hello G., “Derivation of complete crack-tip expansion from Westergaard-Sanfors solutions”, International Journal of Solids and Structures, 144–145 (2019), 265–275 | DOI

[35] Stepanova L., Roslyakov P., Gerasimova T., “Complete Williams Asymptotic Expansion near the Crack Tips of Collinear Cracks of Equal Lengths in an Infinite Plane”, Solid State Phenomena, 258 (2017), 209–212 | DOI

[36] Stepanova L., “Complete Williams Asymptotic Expansion Near The Crack Tips of Collinear Cracks of Equal Lengths in an Infinite Plane Medium.”, Procedia Structural Integrity, 2 (2016), 1789–1796 | DOI

[37] Chona R., Irwin G.R., Shukla A., “Two and three parameter presentations of crack-tip stress field”, The Journal of Strain Analysis for Engineering Design, 17:2 (1982), 79–86 | DOI

[38] Nejati M., Ghouli S., Ayatollahi M.R., “Crack tip asymptotic fields in anisotropic planes: Importance of higher order terms”, Applied Mathetical Modelling, 91 (2021), 837–862 | DOI

[39] Sanchez M., Mallor C., Canales M., Calvo S., Nunez J.L., “Digital Image Correlation parameters optimazed for the characterization of fatigue crack growth life”, Measuraments, 174 (2021), 109082 | DOI

[40] Ayatollahi M.R., Nejati M., Ghouli S., “Crack tip fields in anisotropic planes: a review”, International Journal of Fracture, 2021, 1573–2673 | DOI

[41] Akbardoost Ja., “Scaling effect on the mixed-mode fracture path of rock materials”, Physical Mesomechanics, 19:4 (2016), 441–451 | DOI

[42] Matvienko Yu.G., “Two-parameter fracture mechanics in contemporary strength problems”, Journal of Machinery Manufacture and Reliability, 42:5 (2013), 374–381 | DOI

[43] Bol'shakov A.M., Prokop'ev L.A., “Prediction of a crackh growth trajectory with allowance for the angular distribution of the small components of the tangential stresses at a crack tip”, Russian Metallurgy (Metally), 2019, no. 10, 964–966 (In English; Russian original) | DOI | DOI

[44] Lutsenko A.N., Odintsev I.N., Grinevich A.V., Severov P.B., Plugatar' T.P., “Investigation of the crack propagation process using local strain response measurement data I. Stress Field”, TsAGI Scientific Notes, 46:7 (2015), 55–80 (In Russ.)

[45] Litvinov I.A., Matvienko Yu.G., Razumovsky I.A., “On the accuracy of determination of nonsingular component in stress field at crack tip using extrapolation method”, Mashinostroenie i inzhenernoe obrazovanie, 2014, no. 4(41), 43–51 (In Russ.)

[46] Tyrymov A.A., “Numerical evaluation of T-stress for centrally cracked specimen based on graph model of elastic solid”, Izvesia Volgograd State Technical University, no. 6:229 (2019), 26–29 (In Russ.)

[47] Tyrymov A.A., “Numerical evaluation of stress intensity factor and stress biaxiality for compact specimen based on graph model elastic solid”, Izvesia Volgograd State Technical University, 2020, no. 2(237), 58–61 (In Russ.) | DOI

[48] Matvienko Yu.G., “The non-singular T-stresses in fracture mechanics criteria of solids with nothces”, Vestnik of Lobachevsky University of Nizhni Novgorod, 2011, no. 4–5, 2651–2652 (In Russ.)

[49] Acanfora M., Gallo P., Razavi S.M.J., Ayatollahi M.R., Berto F., “Numerical evaluation of T-stress under mixed mode loading through the use of coarse meshes”, Physical Mesomechanics, 21:1 (2018), 124–134 | DOI

[50] Shlyannikov V.N., Zakharov A.P., Gerasimenko A.A., “Characteristics of cyclic crack resistance of St-3 steel under biaxial loading”, Transactions of Academenergo, 2013, no. 4, 91–101 (In Russ.)

[51] Shlyannikov V.N., Zakharov A.P., “Generalized fatigue fracture diagram under biaxial loading”, Transactions of Academenergo, 2013, no. 4, 72–89 (In Russ.)

[52] Stepanova L.V., Belova O.N., Turkova V.A., “Determination of the Williams series expansion's coefficients using digital photoelasticity method and finite element method”, Vestnik Samarskogo universiteta. Estestvennonauchnaia seriia = Vestnik of Samara University. Natural Science Series, 25:3 (2019), 62–82 (In Russ.) | DOI

[53] Stepanova L.V., Roslyakov P.S., “Complete asymptotic expansion M. Williams near the crack tips of collinear cracks of equal lengths in an infinite plane medium”, PNRPU Mechanics Bulletin, 2015, no. 4, 188–225 | DOI

[54] Mehdi-Soozani A., Experimental fracture mechanics through digital image analysis, Retrospective Theses and Dissertations, Iowa State University, Iowa, 1986, 157 pp. | DOI

[55] Ramesh K., Gupta S., Kelkar A.A., “Evaluation of stress field parameters in fracture mechanics by photoelasticity — revisited”, Engineering Fracture Mechanics, 56:1 (1997), 25–41 ; 43–45 | DOI

[56] Ramesh K., Digital Photoelasticity: Advanced Techniques and Applications, Springer, Berlin, 2000, 410 pp. | DOI

[57] Kosygin A.N., Kosygina L.N., “Digital processing of interferograms obtained by the photoelasticity method”, Vestnik Samarskogo universiteta. Estestvennonauchnaia seriia = Vestnik of Samara University. Natural Science Series, 25:2 (2019), 75–91 (In Russ.) | DOI

[58] Malikova L., Vesely V., “Significance of Higher-order Terms of the Williams Expansion for Plastic Zone Extent Estimation Demonstrated on a Mixed-mode Geometry”, Procedia Materials Science, 3 (2014), 1383–1388 | DOI

[59] Aytaollahi M.R., Moazzami M., “Digital image correlation method for calculating coefficients of Williams expansion in compact specimen”, Optics and Lasers in Engineering, 90 (2017), 26–33 | DOI

[60] Moazzami M., Aytaollahi M.R., Chamani H.R., Guagliano M., Vergani L., “Determination of higher order stress terms in cracked Brazilian disc specimen under mode I loading using digital image correlation technique”, Optics and Laser Technology, 107 (2018), 344–352 | DOI

[61] Su R.K.L., Feng W.J., “Accurate determination of mode I and II leading coefficients of the Williams expansion by finite element analysis”, Finite Elements in Analysis and Design, 41 (2005), 1175–1186 | DOI

[62] Roux-Langlois C., Gravouil A., Baietto M. S.-C., Rethore J., Mathieu F., Hild F., Roux S., “DIC identification and X-FEM simulation of fatigue crack growth based on the Williams series”, International Journal of Solids and Structures, 53 (2015), 38–47 | DOI

[63] Stepanova L.V., “Influence of the higher order terms in Williams' series expansion of the stress field on the stress-strain state in the vicinity of the crack tip. Part I”, Vestnik Samarskogo universiteta. Estestvennonauchnaia seriia = Vestnik of Samara University. Natural Science Series, 25:1 (2019), 63–79 (In Russ.) | DOI

[64] Stepanova L.V., “Influence of the higher order terms in Williams' series expansion of the stress field on the stress-strainstate in the vicinity of the crack tip. Part II”, Vestnik Samarskogo universiteta. Estestvennonauchnaia seriia = Vestnik of Samara University. Natural Science Series, 25:1, 80–96 (In Russ.) | DOI

[65] Malikova L., “Multi-parameter fracture criteria for the estimation of crack propagation direction applied to a mixed-mode geometry”, Engineering Fracture Mechanics, 143 (2015), 32–46 | DOI

[66] Camacho-Reyes A., Vasco-Olmo J.M., James M.N., Diaz F.A., “Towards a new methodology for the characterization of crack tip fields based on a hybrid computational approach”, International Journal of Fatigue, 162 (2022), 106942 | DOI

[67] Mirzaei A.M., Bahrami B., Aytollahi M.R., “Asymptotic stress field around the blunt and sharp notches in bimaterial media under mixed mode I/II loading”, Applied Mathematical Modelling, 109 (2022), 848–863 | DOI

[68] Liu Z.-E., Wei Y., “An analytical solution to the stress fields of kinked cracks”, Journal of the Mechanics and Physics of Solids, 156 (2021), 104619 | DOI

[69] Stepanova L.V., “Asymptotic analysis of the crack tip stress field (consideration of higher order terms)”, Numerical Analysis and Applications, 12:3 (2019), 284–296 (In English; Russian original) | DOI | DOI

[70] Sanford R.J., “A critical re-examination of the Westergaard method for solving opening-mode crack problems”, Mechanics Research Communications, 6 (1979), 289–294 | DOI