The method of solution of the elastic-plastic boundary value problem of tension of strip with~stress raisers with~allowance for~local domains of softening plasticity of material

V. P. Radchenko ; S. V. Gorbunov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 4 (2014), pp. 98-110.

Résumé

The way of solution of the coupled boundary value problem of solid body deformation for the case of a plastically softening material is offered. The strain and stress fields obtained by the simulated undamaged construction behavior modeling under the action of fictitious forces are used as basic data for calculation. The equivalence of simulated undamaged medium strains and real medium strains is supposed. At each point of construction the damage parameter $\omega$ is calculated by means of constitutive relations of the endochronic plasticity theory. This damage parameter associates the components of the true stress tensor $\sigma_{ij}$ of simulated undamaged medium and the engineering stress tensor $\sigma^0_{ij}$ of real medium by $\sigma^0_{ij}=\sigma_{ij}/(1+\omega)$. Using the tensor $\sigma^0_{ij}$ we can calculate the generalized forces of real construction. The problems of tension of the plates weakened with centric circular hole and semicircular notches are solved and the necessary experiments are conducted. The strain and true stress fields are obtained by numerical calculation at the finite element analysis software and are used for the engineering stress of real construction computation according to the foregoing expression. Softening plasticity domains are plotted. It is found that at the moment before failure the stage of post critical deformation is implementing in the region of stress concentration, although the curve “total displacement – axial force” corresponds to the stage of plastic hardening.

Détail

Keywords: boundary value problem, stress raiser, post critical deformation, plane stress state, simulated damaged medium, plasticity, experimental proof.

Bibliographie

[1] Bylia O. I., Vasin R. A., “Behavior of alloys in superplastic and near to superplastics regimes of deforming”, Izvestiia Tul'skogo gosudarstvennogo universiteta. Estestvennye nauki, 2011, no. 2, 116–127 (In Russian)

[2] Vasin R. A., Enikeev F. U., Mazurskii M. I., “Materials with a Falling Diagram”, Mech. Solids, 1995, no. 2, 168–169 | MR

[3] Lebedev A. A., Chausov N. G., Marusii O. I., Evetskii Yu. L., Grishai G. Kh., Grinenko B. G., “Failure kinetics for sheet austenitic steels in the final stage of deformation”, Strength of Materials, 21:3 (1989), 285–290 | DOI

[4] Mironov V. I., “Properties of the material in rheologically unstable state”, Zavodskaia Laboratoriia. Diagnostika Materialov, 68:10 (2002), 47–52 (In Russian)

[5] Struzhanov V. V., Mironov V. I., Deformational Softening of Material in Structural Elements, Ural Branch of Russian Academy of Sciences, Ekaterinburg, 1995, 191 pp. (In Russian)

[6] Chausov N. G., Zasimchuk U. E., Markashova L. I., Vil'deman V. E., Turchak T. V., Pilipenko A. P., Parats V. M., “Features of plastic deformation of materials under dynamic non-equilibrium processes”, Zavodskaia Laboratoriia. Diagnostika Materialov, 2009, no. 6, 52–59 (In Russian)

[7] Faleskog J., Barsoum I., “Tension-torsion fracture experiments. Part I: Experiments and a procedure to evaluate the equivalent plastic strain”, International Journal of Solids and Structures, 50:25–26 (2013), 4241–4257 | DOI

[8] Vil'demann V. E., Tretyakov M. P., “Material testing by plotting total deformation curves”, J. Mach. Manuf. Reliab., 42:2 (2013), 166–170 | DOI

[9] Wildemann V. E., Lomakin E. V., Tretyakov M. P., “Postcritical deformation of steels in plane stress state”, Mech. Solids, 49:1 (2014), 18–26 | DOI

[10] Zhizherin S. V., Struzhanov V. V., Mironov V. I., “Iterative methods of stress calculation in pure bending of beams of damageable material”, Vychisl. Tekhnol., 6:5 (2001), 24–33 (In Russian) | Zbl

[11] Struzhanov V. V., “Determination of incremental modules on base of tension test results with full diagram construction”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2008, no. 1(16), 160–163 (In Russian) | DOI

[12] Struzhanov V. V., “The properties of softening materials and constitutive relations under uniaxial stress state”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2007, no. 2(15), 69–78 (In Russian) | DOI

[13] Vil'deman V. E., Il'inykh A. V., “Simulation of structural failure and scale effects of softening at the post-critical deformation stage in heterogeneous media”, Fiz. Mezomekh., 10:4 (2007), 23–29 (In Russian)

[14] Ilinykh A. V., “Numerical modeling of structural fracture processes for granular composites with isotropic elements of structure”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2011, no. 2(23), 101–106 (In Russian) | DOI

[15] Andreeva E. A., “Solution of one-dimensional softening materials plasticity problems”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2008, no. 2(17), 152–160 (In Russian) | DOI

[16] Gorbunov S. V., “The way of solution of the boundary value problem of a pure bending of a beam from plastically softening material”, Proceedings of the Ninth All-Russian Scientific Conference with international participation. Part 1, Matem. Mod. Kraev. Zadachi, Samara State Technical Univ., Samara, 2013, 92–96 (In Russian)

[17] Radchenko V. P., Eremin Iu. A., Reologicheskoe deformirovanie i razrushenie materialov i elementov konstruktsii [Rheological Deformation and Failure of Materials and Structural Elements], Mashinostroenie-1, Moscow, 2004, 264 pp. (In Russian)

[18] Lemaitre J., “Evolution of dissipation and damage in metals submitted to dynamic loading”, Proc. I.C.M. (Kyoto, Japan), v. I, 1971, 151–157

[19] Lemaitre J., Chaboche J. L., “Aspect phénoménologique de la rupture par endommagement”, J. de Mécanique Appliqué, 2:3 (1978), 317–365 (In French)

[20] Rabotnov Yu. N., Polzuchest' elementov konstruktsii [Creep of Structural Elements], Nauka, Moscow, 1966, 752 pp. (In Russian)

[21] Adeyanov I. E., Klebanov Ya. M., “Influence of the material damage effect on the softening factor”, Proceedings of the Sixth All-Russian Scientific Conference with international participation (1–4 June 2009). Part 1, Matem. Mod. Kraev. Zadachi, Samara State Technical Univ., Samara, 2009, 10–12 (In Russian)