Geodesic
Potential Fields of Free Energy at the Stages of Hardening and Softening of the Hencky Medium at Nonpositivity of Volume Deformation
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 2 (2014), pp. 82-88.

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The Hencky medium with softening under isothermal and quasistatic deformation is considered. It is believed that volume deformation is not positive. In this case softening is characterized by the part of union curve with negative slope. For aforementioned conditions function of free energy is presented. For all stages of deformation in the space “volume deformation – intensity of shear's deformation” level lines of the free energy are constructed. It is established that level lines are ellipses in hardening and function of free energy increases with distance from their centers, while in softening hyperbolas are level lines and function of free energy decreases with distance from their centers. Obtained results indirectly confirm the change in type of boundary value problem from elliptic to hyperbolic under material transition to the softening.
Keywords: Hencky medium, free energy, potential fields, level lines, hardening, softening. 
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K. V. Berdnikov; V. V. Struzhanov. Potential Fields of Free Energy at the Stages of Hardening and Softening of the Hencky Medium at Nonpositivity of Volume Deformation. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 2 (2014), pp. 82-88. http://geodesic.mathdoc.fr/item/VSGTU_2014_2_a6/

[1] V. A. Ibragimov, V. D. Klyushnikov, “Some problems for media with falling stress-strain diagram”, Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 1971, no. 4, 116–121 (In Russian)

[2] A. A. Lebedev, N. G. Chausov, “Phenomenological fundamentals of the evaluation of crack resistance of materials on the basis of parameters of falling portions of strain diagrams”, Strength of Materials, 15:2 (1983), 155–160 | DOI

[3] V. V. Struzhanov, Vyach. V. Bashurov, “The modified Masing model”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2007, no. 1(14), 29–39 (In Russian) | DOI

[4] V. P. Radchenko, E. V. Nebogina, M. V. Basov, “Structural-phenomenological approach to the description of a complete diagram of elastic-plastic deformation”, Izv. vuzov. Mashinostroenie, 2000, no. 5–6, 3–13 (In Russian)

[5] Yu. I. Kadashevich, S. P. Pomytkin, “Investigation of uniaxial and biaxial loadings of softening materials in endochronic theory of inelasticity”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2012, no. 1(26), 110–115 (In Russian) | DOI

[6] V. V. Struzhanov, “Associated and incremental laws of plastic flow for mediums exhibiting strain-softening behaviour”, Izvestiya Ural'skogo gosudarstvennogo universiteta, 1998, no. 10, 92–101 (In Russian) | Zbl

[7] N. G. Chausov, “Complete stress-strain diagram as a source of information on the kinetics of damage accumulation and crack resistance of materials”, Zavodskaya laboratoriya. Diagnostika materialov, 2004, no. 7, 42–49 (In Russian)

[8] B. V. Gorev, I. A. Banshchikova, “To the description of softening stage of “stress–strain” diagram with scalar damage parameter kinetic equations”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2008, no. 2(17), 110–117 (In Russian) | DOI

[9] V. E. Vil'deman, M. P. Tretyakov, “Material testing by plotting total deformation curves”, Journal of Machinery Manufacture and Reliability, 42:2 (2013), 166–170 | DOI

[10] V. E. Vil'deman, M. P. Tretyakov, “Analysis of the effect of loading system rigidity on postcritical material strain”, Journal of Machinery Manufacture and Reliability, 42:3 (2013), 219–226 | DOI

[11] V. V. Struzhanov, E. A. Bakhareva, “Mathematical methods in the theory of pure bending of rectangular beams made of weakening material with symmetric stress-strain diagram”, Computational continuum mechanics, 5:2 (2012), 158–167 (In Russian) | DOI

[12] V. V. Struzhanov, V. I. Mironov, Deformatsionnoye razuprochneniye materiala v elementakh konstruktsiy [Strain softening of material in structural elements], Publishing House of UB RAS, Ekaterinburg, 1995, 192 pp. (In Russian)

[13] L. V. Nikitin, E. I. Ryzhak, “Feasibility of realizing states of material corresponding to ‘falling’ section of diagram”, Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 1986, no. 2, 155–161 (In Russian)

[14] E. I. Ryzhak, “To the Problem of Realizability of Homogeneous Postcritical Deformation in a Rigid Triaxial Testing Machine”, Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 1991, no. 1, 111–127 (In Russian)

[15] V. P. Radchenko, E. V. Nebogina, E. A. Andreeva, “Structural Model of Material Softening at Creep under Complex Stress Conditions”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2009, no. 1(18), 75–84 (In Russian) | DOI

[16] V. V. Struzhanov, K. V. Berdnikov, “On defining relations for the Hencky environment of softening of the material under diagonal stress tensor”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2012, no. 3(28), 72–80 (In Russian) | DOI | Zbl

[17] A. I. Lurie, Theory of Elasticity, Springer, Berlin, Heidelberg, New York, 2005, 1050 pp. | DOI | Zbl