Tensor theory of deformation damage

N. D. Tutyshkin ; V. Yu. Travin

Čebyševskij sbornik, Tome 23 (2022) no. 5, pp. 320-336.

Résumé

On the basis of the physical concept of pore formation, origin and growth of pores, generalized determining relations of the tensor model of plastic damage of metals based on three invariants are formulated. The multiplicative decomposition of the metric transform tensor and the thermodynamic formulation of the defining relations lead to a symmetric damage tensor of the second rank with a clear physical meaning. Its first invariant determines the damage associated with the plastic dilatance of the material due to pore growth, the second invariant of the deviant tensor - damage associated with a change in the shape of defects, the third invariant of the deviant tensor describes the effect on the damage of the type of stress state (Lode angle), including the effect of the rotation of the main axes of the stress tensor (change of the Lode angle). The introduction of three component measures with the corresponding physical meaning allows the kinetic process of deformation damage to be represented by an equivalent parameter in a three-dimensional vector space, including the criterion conditions for plastic destruction. A measure of plastic damage based on three invariants can be useful in assessing the quality of the mesostructure of metal products obtained by pressure treatment methods.

Détail

Keywords: basic equations, defining relation, plasticity, stresses, strains, physical and structural parameters, damage, energy dissipation, loading surface.

Bibliographie

[1] CityTutyshkin N.D., Müller W.H., Wille R., Zapara M.A., “Strain-induced damage of metals under large plastic deformation: Theoretical framework and experiments”, International Journal of Plastisity, 59 (2014), 133–151

[2] Tutyshkin N.D., Lofink P., Mller W.H., Wille R., Stahn O., “Constitutive equations of a tensorial model for strain-induced damage of metals based on three invariants”, International Journal Continuum Mechanics and Thermodynamics, 29:1 (2017), 251–269

[3] Brnig M., “An anisotropic ductile damage model based on irreversible thermodynamics”, International Journal of Plasticity, 19 (2003), 1679–1713

[4] Bammann D.J., Solanki K.N., “On kinematic, thermodynamic, and kinetic coupling of a damage theory for polycrystalline material”, International Journal of Plasticity, 26 (2010), 775–793

[5] Tutyshkin N. D., Tregubov V. I., Svyazannye zadachi teorii povrezhdaemosti deformiruemykh materialov, ed. N.D. Tutyshkin, TulGU–RARAN, Tula, 2016, 267 pp.

[6] Bogatov A.A., Mizhiritskii O.I., Smirnov S.V., Resurs plastichnosti metallov pri obrabotke davleniem, Metallurgiya, M., 1984, 144 pp.

[7] Bao Y., Wierzbicki T., “On fracture locus in the equivalent strain and stress triaxiality space”, International Journal of Mechanical Sciences, 46 (2004), 81–98

[8] Bao Y., Wierzbicki T., “On the cut-off value of negative triaxiality for fracture”, Journal Engineering. Fracture. Mechanics, 72 (2005), 1049–1069

[9] Xue L., “Damage accumulation and fracture initiation of uncracked ductile solids subjected to triaxial loading”, International Journal of Solids and Structures, 44 (2007), 5163–5181

[10] Dunand M., Maertens A. P., Luo M., Mohr D., “Experiments and modeling of anisotropic aluminum extrusions under multi-axial loading – Part I: Plasticity”, International Journal of Plasticity, 36 (2012), 34–49

[11] Luo M., Dunand M., Mohr D., “Experiments and modeling of anisotropic aluminum extrusions under multi-axial loading – Part II: Ductile fracture”, International Journal of Plasticity, 32–33 (2012), 36–58

[12] Khan A.S., Liu H., “A new approach for ductile fracture prediction on Al 2024-T351 alloy”, International Journal of Plasticity, 35 (2012), 1–12

[13] Brnig M., Gerke S., Hagenbrock V., “Micro-mechanical studies on the effect of the stress triaxiality and the Lode parameter on ductile damage”, International Journal of Plasticity, 5 (2013), 49–65

[14] Danas K., Ponte Castaseda P., “Influence of the Lode parameter and the stress triaxiality on the failure of elasto-plastic porous materials”, International Journal of Plasticity, 49 (2012), 1325–1342

[15] Hosokava A., Wilkinson D. S., Kang J., Maire E., “Onset of void coalescence in uniaxial tension studied by continuous X-ray tomography”, International Journal Acta Materialia, 61 (2013), 1021–1036

[16] Khill R., Matematicheskaya teoriya plastichnosti, Per. s angl. E.I. Grigolyuka, Gosud. izd-vo tekhniko-teoret. lit-ry, M., 1956, 407 pp.

[17] Sokolovskii V.V., Teoriya plastichnosti, 3-e izd., pererab. i dop., Vysshaya shkola, M., 1969, 608 pp.

[18] Kachanov L.M., Osnovy teorii plastichnosti, Nauka, M., 1969, 420 pp.

[19] Ivlev D.D., Teoriya idealnoi plastichnosti, Nauka, M., 1966, 232 pp.

[20] Sedov L. I., Mekhanika sploshnoi sredy, V 2 t., v. 1, 4-e izd., ispravl. i dop., Nauka, M., 1984, 528 pp.

[21] Zapara M.A., Tutyshkin N.D., Müller W.H., Wille R., “Constitutive equations of a tensorial model for ductile damage of metals”, International Journal Continuum Mechanics and Thermodynamics, 24 (2012), 697–717

[22] Zapara M.A., Tutyshkin N.D., Müller W.H., Wille R., “A study of ductile damage and failure of pure copper – Part II: Analysis of the deep drawing process of a cylindrical shell”, Journal of Technische Mechanik, 32 (2012), 631–648

[23] Benzerga A., Surovik D., Keralavarma S., “On the path-dependence of the fracture locus in ductile materials – Analysis”, International Journal of Plasticity, 37 (2012), 157–170

[24] Rabotnov Yu.N., Vvedenie v mekhaniku razrusheniya, Nauka, M., 1987, 80 pp.

[25] Green R.J., “A plasticity theory for porous solids”, International Journal of Mechanical Sciences, 14 (1972), 215–224

[26] Ekobori T., Fizika i mekhanika razrusheniya i prochnosti tverdykh tel, Metallurgiya, M., 1971, 264 pp.