On the determination of specific elastic energy flow to the vertex of a physical cut via a finite element solution

V. V. Glagolev   ; A. I. Lutkhov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 90 (2024), pp. 78-89

Résumé

The finite element approximation of a double cantilever beam (DCB) specimen with a physical cut in a linear elastic medium is considered. The thickness of the physical cut specifies a linear parameter of the problem. The $J$-integral is determined as the product of the linear parameter and the average value of the specific elastic energy on the dead-end edge of the finite element. For the considered loading schemes of the DCB specimen in modes I and II with zero linear parameter set in ANSYS, the stress intensity factors are obtained and used to determine the $J$-integrals. The convergence of the product of the linear parameter and the average value of the specific elastic energy on the dead-end edge of the finite element to the reference values of the $J$-integrals is shown for equivalent loading of the specimen with a physical cut and with a linear parameter tending to zero. The specific work of nodal forces is studied during the dead-end finite element removing. The convergence of the specific work of nodal forces when removing the dead-end element by simple unloading of adjacent edges to the value of the reference $J$-integral is observed.

Détail

Keywords: stress intensity factor, mathematical cut, physical cut, elastic energy flow, finite element method, linear parameter, $J$-integral, Neuber-Novozhilov approach.

Bibliographie

[1] Griffith A.A., “The phenomena of rupture and flow in solids”, Philosophical Transactions of the Royal Society of London. Ser. A, 221 (1921), 163–189 | DOI

[2] Cherepanov G.P., Mekhanika khrupkogo razrusheniya, Nauka, M., 1974, 640 pp.

[3] Murakami Yu., Spravochnik po koeffitsientam intensivnosti napryazhenii, Mir, M., 1990, 1014 pp.

[4] Malik A.V., Lavit I.M., “Metod rascheta koeffitsienta intensivnosti napryazhenii dlya nepodvizhnoi treschiny normalnogo razryva pri dinamicheskom nagruzhenii”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2018, no. 54, 88–102 | Zbl

[5] Murakami Y., “A simple procedure for the accurate determination of stress intensity factors by finite element method”, Engineering Fracture Mechanics, 8:4 (1976), 643–655 | DOI

[6] Rybicki E.F., Kanninen M.F., “A finite element calculation of stress intensity factors by a modi fied crack closure integral”, Engineering Fracture Mechanics, 9:4 (1977), 931–938 | DOI

[7] Caicedo J., Portela A., “Direct computation of stress intensity factors in finite element method”, European Journal of Computational Mechanics, 26:3 (2017), 309–335 | DOI | MR

[8] Tanaka M., Hamada M., Iwata Y., “Computation of a two-dimensional stress intensity factor by the boundary element method”, Ingenieur-Archiv, 52 (1982), 95–104 | DOI | Zbl

[9] Cherepanov G.P., “Some new applications of the invariant integrals of mechanics”, Journal of Applied Mathematics and Mechanics, 76:5 (2012), 519–536 | DOI | MR

[10] Rice J.R., “A path independent integral and the approximate analysis of strain concentration by notches and cracks”, Journal of Applied Mechanics, 35:2 (1968), 379–386 | DOI

[11] Kolednik O., Schongrundner R., Fischer F.D., “A new view on J-integrals in elastic-plastic materials”, International Journal of Fracture, 187:1 (2014), 77–107 | DOI

[12] Huang K., Shimada T., Ozaki N., Hagiwara Y., Sumigawa T., Guo L., Kitamura T., “A unified and universal Griffith-based criterion for brittle fracture”, International Journal of Solids and Structures, 128 (2017), 67–72 | DOI

[13] Prandtl L., Knauss W.G., “A thought model for the fracture of brittle solids”, International Journal of Fracture, 171:2 (2011), 105–109 | DOI

[14] Entov V.M., Salganik R.L., “K modeli khrupkogo razrusheniya Prandtlya”, Izvestiya AN SSSR. Mekhanika tverdogo tela, 1968, no. 6, 87–99

[15] He X., “A review of finite element analysis of adhesively bonded joints”, International Journal of Adhesion and Adhesives, 31:4 (2011), 248–264 | DOI

[16] Novozhilov V.V., “O neobkhodimom i dostatochnom kriterii khrupkoi prochnosti”, Prikladnaya matematika i mekhanika, 33:2 (1969), 212–222

[17] Morozov N.F., Semenov B.N., “Primenenie kriteriya khrupkogo razrusheniya V.V. Novozhilova pri opredelenii razrushayuschikh nagruzok dlya uglovykh vyrezov v usloviyakh slozhnogo napryazhennogo sostoyaniya”, Izvestiya AN SSSR. Mekhanika tverdogo tela, 1986, no. 1, 122–126

[18] Neiber G., Kontsentratsiya napryazhenii, OGIZ: Gostekhizdat, M.–L., 1947, 204 pp.

[19] Neuber H., Theory of Notch Stresses: Principles for Exact Calculation of Strength with Reference to Structural Form and Material, Springer-Verlag, Berlin, 1958, 180 pp.

[20] Nazarov S.A., Paukshto M.V., Diskretnye modeli i osrednenie v zadachakh teorii uprugosti, Izd-vo Leningr. un-ta, L., 1984, 93 pp.

[21] Morozov E.M., ANSYS v rukakh inzhenera: mekhanika razrusheniya, Lenand, M., 2010, 456 pp.

[22] Burtsev A.Yu., Glagolev V.V., Markin A.A., “Issledovanie protsessa lokalnoi razgruzki elementa v konechno-elementnom kontinuume”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2021, no. 69, 86–96

[23] Berto F., Glagolev V., “V, Markin A.A. Relationship between Jc and the dissipation energy in the adhesive layer of a layered composite”, International Journal of Fracture, 224 (2020), 277–284 | DOI

[24] Andrews M.G., Massabo R., “The effects of shear and near tip deformations on energy release rate and mode mixity of edgecracked orthotropic layers”, Engineering Fracture Mechanics, 74 (2007), 2700–2720 | DOI