On account of viscous properties of materials in the theory of large elastoplastic strains

S. V. Belykh ; A. A. Burenin ; L. V. Kovtanyuk ; A. N. Prokudin

Čebyševskij sbornik, Tome 18 (2017) no. 3, pp. 108-130.

Résumé

A geometrically and thermodynamically consistent mathematical model of large strains of materials with elastic, viscous and plastic properties is proposed. It is believed that at the stage of a strain, which precedes the plastic flow and during unloading, the viscous material properties provide the creep process and thus a slow growth of irreversible strains. While rapid growth of irreversible strains under plastic flow conditions, viscous properties act as a mechanism that retards the flow. The accumulation of irreversible strains, therefore, occurs successively: initially, in the creep process, then under plastic flow and, finally, again due to creep of the material (during unloading). On the elastoplastic boundaries advancing along the deformable material, there is a change in the growth mechanism of irreversible strains from creep to plasticity and vice versa. Such a change is possible only under conditions of continuity of irreversible strains and their change rates, which imposes the requirement of consistency in the definitions of irreversible stress distribution rates, i.e., the laws of creep and plasticity. Changing the production mechanisms of irreversible strains means various setting up of the source in the differential equation of the change (transfer) of these strains, hence irreversible strains are not divided into plastic strains and creep strains. To maximize the visibility of the model's correlations, the hypothesis on the independence of thermodynamic potentials (internal energy, free energy) on irreversible strains is accepted. As a consequence of the hypothesis, an analog of the Murnaghan formula is obtained, the classical position of the elastoplasticity is that the stresses in the material are completely determined by the level and distribution of reversible strains. The main provisions of the proposed model are illustrated by the solution in its framework of the boundary value problem of the elastoviscoplastic material motion in a pipe due to a varying pressure drop.

Détail

Keywords: large deformations, elasticity, viscoelasticity, plasticity, unloading.

Bibliographie

[1] Olejnikov A. I., Pekarsh A. I., Integrated design of monolithic panel manufacturing processes, Ekom, M., 2009

[2] Hill R., “A general theory of uniqueness and stability in elastic-plastic solids”, Journal of the Mechanics and Physics of Solids, 6:3 (1958), 236–249 | DOI | MR | Zbl

[3] Truesdell C., “Hypo-elasticity”, J. Rat. Mech. Anal., 4 (1955), 83–133 | MR | Zbl

[4] Simo J. C., Pister K. S., “Remarks on rate constitutive equations for finite deformation problems: computational implications”, Computer Methods in Applied Mechanics and Engineering, 46:2 (1984), 201–215 | DOI | Zbl

[5] Khan A. S., Huang S. J., Continuum Theory of Plasticity, Wiley, New York, 1995 | Zbl

[6] Xiao H., Bruhns O.T., Meyers A., “Elastoplasticity beyond small deformations”, Acta Mech., 182 (2006), 31–111 | DOI | Zbl

[7] Firat M., Kaftanoglu B., Eser O., “Sheet metal forming analyses with an emphasis on the springback deformation”, J. Mater. Process. Technol., 196:1–3 (2008), 135–148 | DOI

[8] Lee E. H., “Elastic-Plastic Deformation at Finite Strains”, ASME. J. Appl. Mech., 36:1 (1969), 1–6 | DOI | Zbl

[9] Naghdi P. M., “A critical review of the state of finite plasticity”, ZAMP, 41 (1990), 315–394 | DOI | MR | Zbl

[10] Vladimirov I.N., Pietryga M.P., Reese S., “Anisotropic finite elastoplasticity with nonlinear kinematic and isotropic hardening and application to sheet metal forming”, Int. J. Plasticity, 26:5 (2010), 659–687 | DOI | Zbl

[11] Sansour C., Karšaj I., Soric J., “On a numerical implementation of a formulation of anisotropic continuum elastoplasticity at finite strains”, J. Comput. Phys., 227:16 (2008), 7643–7663 | DOI | MR | Zbl

[12] Levin V. I., Multiple superposition of large deformations in elastic and viscoelastic bodies, Nauka. Fizmatlit, M., 1999, 223 pp.

[13] Levin V. I., “The theory of multiple superposition of large deformations and its industrial implementation in a full-featured CAE for strength engineering analysis”, Izvestiya Tul'skogo gosudarstvennogo universiteta. Yestestvennyye nauki, 2013, no. 2-2, 156–178

[14] Markin A. A., Sokolova M. Yu., Thermomechanics of elastoplastic deformation, Fizmatlit, M., 2013, 319 pp.

[15] Pozdeyev A. A, Trusov P. V., Nyashin Yu. I., Large elastic-plastic deformations: theory, algorithms, applications, Nauka, M., 1986, 232 pp.

[16] Burenin A.A., Bykovtsev G.I., Kovtanyuk L.V., “One simple model for elastoplastic medium at finite deformations”, Doklady akademii nauk, 347:2 (1996), 199–201

[17] Burenin A. A., Kovtanyuk L. V., Large irreversible deformations and elastic aftereffects, Dal'nauka, Vladivostok, 2013, 312 pp.

[18] Burenin A.A., Kovtanyuk L.V., Polonik M.V., “Forming a one-dimensional residual stress field in the vicinity of a cylindrical defect of continuity of an elastoplastic medium”, Prikladnaya matematika i mekhanika, 64:2 (2003), 316–325

[19] Burenin A. A., Ustinova A. S., “Development and inhibition of viscoelastic viscous flow with the calculation of elastic response after stopping flow and unloading”, Progress in Continuum Mechanics, To the 70th anniversary of V. A. Levin, Vladivostok, 2009, 91–102

[20] Kovtanyuk L.V., “On extrusion of elastoviscoplastic material through a rigid cylindrical matrix”, Doklady akademii nauk, 400:6 (2005), 764–767

[21] Kovtanyuk L. V., “Viscoplastic flow and residual stresses in a heavy layer of incompressible material located on an inclined plane”, Mathematical models and methods of continuum mechanics, To the 60th anniversary of A. A. Burenin, Vladivostok, 2007, 120–128

[22] Burenin A.A., Kovtanyuk L.V., Mazelis A. L., “Development and inhibition of a rectilinear axisymmetric viscoplastic flow and elastic aftereffect after its stoppage”, Prikladnaya mekhanika i tekhnicheskaya fizika, 51:2 (2010), 140–147

[23] Kovtanyuk L.V., “Modeling of large elastoplastic deformations in the non-isothermal case”, Dal'nevostochnyy matematicheskiy zhurnal, 5:1 (2004), 104–117

[24] Burenin A.A., Kovtanyuk L.V., Panchenko G. L., “Nonisothermal motion of an elastoviscoplastic medium through a pipe under a changing pressure drop”, Doklady akademii nauk, 464:3 (2015), 284–287

[25] Bykovtsev G. I., Ivlev D. D., Theory of plasticity, Dal'nauka, Vladivostok, 1998

[26] Galin L. A., Elastic-plastic problems, Nauka, M., 1984

[27] De Groot S. R., Mazur P., Non-Equilibrium Thermodynamics, North-Holland, 1969 | MR

[28] Bykovtsev G. I., Yarushina V. M., “On the features of the unsteady creep model based on the use of piecewise linear potentials”, Problems of mechanics of continuous media and structural elements, To the 60th anniversary of G.I. Bykovtsev, Dal'nauka, Vladivostok, 1998, 9–26

[29] Burenin A.A., Yarushina V.M., “Plane stress state under nonlinear unsteady creep”, Dal'nevostochnyy matematicheskiy zhurnal, 3:1 (2002), 64–78

[30] Lurie A. I., Nonlinear theory of elasticity, Nauka, M., 1980