2016-Sustainable Industrial Processing Summit
SIPS 2016 Volume 7: Yang Intl. Symp. / Multiscale Material Mechanics

Editors:Kongoli F, Aifantis E, Wang H, Zhu T
Publisher:Flogen Star OUTREACH
Publication Year:2016
Pages:190 pages
ISSN:2291-1227 (Metals and Materials Processing in a Clean Environment Series)
CD shopping page

    Analytical Periodic Shear Band Solutions in Gradient Plasticity

    Hang Xu1; Elias Aifantis2; Ammarah Raees3; Qing Kai Zhao1;
    1SHANGHAI JIAO TONG UNIVERSITY, Shanghai, China; 2ARISTOTLE UNIVERSITY OF THESSALONIKI, Thessaloniki, Greece; 3, shanghai, China;
    Type of Paper: Regular
    Id Paper: 385
    Topic: 1


    The analytical shear band-type solutions are obtained at different periods for steady- state softening and hardening materials for the finite domain. For this purpose, strain gradient plasticity theory is discussed in detail and the constitutive equation for the gradient plasticity is solved using the analytic technique, i.e. homotopy analysis method (HAM), which is also constructed step by step, analyzed and implemented. The nonlinear governing partial differential equation is reduced to the non-dimensionalized nonlinear ordinary differential equation by using the appropriate similarity transformations. Convergent solutions are obtained with the help of optimal convergence-control parameter. Moreover, the error analysis has been performed to guarantee the convergence of our series solution. This present contribution addresses that HAM is a powerful tool for solving a complicated nonlinear problem and it deserves to be applied for more problems in deformation patterning phenomenon.


    Fractional; Plasticity;


    [1] E.C. Aifantis, On the microstructural origin of certain inelastic models, Trans. ASME, J. Eng.Mat.Techn. 106 (1984) 326–330.
    [2] E.C. Aifantis, The physics of plastic deformation, Int. J. Plasticity 3 (1987) 211-247.
    [3] N. Triantafyllidis and E.C. Aifantis, A gradient approach to localization of deformation-I. Hy-perelastic materials, J. Elasticity 16 (1986) 225-238.
    [4] H.M. Zbib and E.C. Aifantis, On the localization and postlocalization behavior of plastic de-formation, I,II and III, Res. Mechanica, Int. J. Struct. Mech. Mater. Sci. 23 (1988) 261 - 277, 279 – 292 and 293 - 305.
    [5] H.M. Zbib and E.C. Aifantis, On the structure and width of shear bands, Scripta Metall. 22 (1988) 703-708.
    [6] H.M. Zbib and E.C. Aifantis, A gradient-dependent model for the Portevien-Le Chatelier ef-fect, Scripta Metall. 22 (1988) 1331–1336.
    [7] H.M. Zbib and E.C. Aifantis, A gradient-dependent flow theory of plasticity: application to metal and soil instabilitie, Appl. Mech. Rev. 42, (2) (1989) 295-304.
    [8] I.Vardoulakis and E.C. Aifantis, A gradient flow theory of plasticity for granular materials, Acta Mech. 87 (1991) 197 217.
    [9] H.B. Muhlhaus and E.C. Aifantis, The influence of microstructure-induced gradients on the localization of deformation in viscoplastic materials, Acta Mech. 89 (1991) 217–231.
    [10] N.C. Charalambakis and E.C. Aifantis, On stress controlled thermoviscoplastic shearing and higher order strain gradients, Acta Mech. 81 (1990) 109-114.
    [11] N.C. Charalambakis, A. Rigatos and E.C. Aifantis, The stabilizing role of higher-order strain gradients in nonlinear thermoviscoplasticity, Acta Mech. 86 (1991) 65–81.
    [12] F. Oka, A. Yashima, T. Adachi, and E.C. Aifantis, A gradient dependent viscoplastic model for clay and its application to FEM consolidation analysis. In: Constitutive laws for engineering materials - Theory and applications, (Desai, C. S., ed.), (1991) 313-316.
    [13] E. C. Aifantis, Pattern formation in plasticity, Int. J. Engng Sci. 33 (1995) 2161-2178.
    [14] E. C. Aifantis, Non-linearity, periodicity and patterning in plasticity and fracture, In.J. Non-linear Mech. 31 (1996) 797-809.
    [15] N.A. Fleck and J.W. Hutchinson A reformulation of strain gradient plasticity, J. Mech. Phys.Solids 49 (2001)22452271.
    [16] M.E Gurtin and L.Anand, Thermodynamics applied to gradient theories: The theories of Aifantis and Hutchinson and their generalization, JMPS 57 (2009) 405-421.
    [17] E. C. Aifantis, On scale invariance in anisotropic plasticity, gradient plasticity and gradient elasticity, Int. J. Engng Sci. 47 (2009) 1089-1099.
    [18] D.Lasry and T.Belytschko, Localization limiters in transient problems, Int. J. Solids Struct. 24 (1988) 581597.
    [19] R. B.Joshi, A. E. Bayoumi, and H. M.Zbib, Evaluation of macroscopic shear banding using a digital image processing technique, Scripta Met. et. Mater. 24 (1990) 1747-1752.
    [20] G.Z. Voyiadjis and R.K. Abu Al-Rub, Gradient plasticity theroy with a variable length scale parameter, Int. J. Solids Struct. 42 (2005) 3998-4029.
    [21] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, 2003, Chapman & Hall/CRC, Boca Raton.
    [22] S.J. Liao, Homotopy analysis method in nonlinear differential equations, 2012, Higher Education Press, Beijing.
    [23] S.J. Liao, An optimal homotopy-analysis approach for strongly nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simulat., 15 (2010) 2003-2016.
    [24] E. C. Aifantis and J.B. Serrin, The mechanical theory of fluid interfaces and Maxwell’s rule, J.Colloid Interf. Sci. 96 (1983) 517-529.
    [25] S.J. Liao, A kind of approximate solution technique which does not depend upon small parameters (II)-An application in fluid mechanics, Int. J. Nonlin. Mech., 32 (1997) 815-822.

    Full Text:

    Click here to access the Full Text

    Cite this article as:

    Xu H, Aifantis E, Raees A, Zhao Q. Analytical Periodic Shear Band Solutions in Gradient Plasticity. In: Kongoli F, Aifantis E, Wang H, Zhu T, editors. Sustainable Industrial Processing Summit SIPS 2016 Volume 7: Yang Intl. Symp. / Multiscale Material Mechanics. Volume 7. Montreal(Canada): FLOGEN Star Outreach. 2016. p. 147-152.