Skip to main content
SpringerLink
Log in

Implicit Nonlocality in the Framework of Viscoplasticity

  • Reference work entry
  • First Online:
Handbook of Nonlocal Continuum Mechanics for Materials and Structures

  • 1605 Accesses

Abstract

The considerations are addressed to the notion of implicit nonlocality in mechanical models. The term implicit means that there is no direct measure of nonlocal action in a model (like classical or fractional gradients, etc. in explicit nonlocal models), but some phenomenological material parameters can be interpreted as one that maps some experimentally observed phenomena responsible for the scale effects.

The overall discussion is conducted in the framework of the Perzyna Theory of Viscoplasticity where the role of the implicit length scale parameter plays the relaxation time of the mechanical disturbance. In this sense, in the viscoplastic range of the material behavior, the deformation at each material point contributes to the finite surrounding. The important consequence is that the solution of the IBVP described by Perzyna’s theory is unique – the relaxation time is the regularizing parameter.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 799.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 999.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • R. Abraham, J.E. Marsden, T. Ratiu, Manifolds, Tensor Analysis and Applications (Springer, Berlin, 1988)

    Book  MATH  Google Scholar 

  • Abaqus, Abaqus Version 6.12 Collection (SIMULIA Worldwide Headquarters, Providence, 2012)

    Google Scholar 

  • E.C. Aifantis, Strain gradient interpretation of size effects. Int. J. Fract. 95, 299–314 (1999)

    Article  Google Scholar 

  • R.J. Asaro, Crystal plasticity. J. Appl. Mech. 50, 921–934 (1983)

    Article  MATH  Google Scholar 

  • T.W. Barbee, L. Seaman, R. Crewdson, D. Curran, Dynamic fracture criteria for ductile and brittle metals. J. Mater. 7, 393–401 (1972)

    Article  Google Scholar 

  • X. Boidin, P. Chevrier, J.R. Klepaczko, H. Sabar, Identification of damage mechanism and validation of a fracture model based on mesoscale approach in spalling of titanium alloy. Int. J. Solids Struct. 43(14–15), 4029–4630 (2006)

    MATH  Google Scholar 

  • D.R. Curran, L. Seaman, D.A. Shockey, Dynamic failure of solids. Phys. Rep. 147(5–6), 253–388 (1987)

    Article  Google Scholar 

  • S. Cochran, D. Banner, Spall studies in uranium. J. Appl. Phys. 48(7), 2729–2737 (1988)

    Article  Google Scholar 

  • R. de Borst, J. Pamin, Some novel developments in finite element procedures for gradient-dependent plasticity. Int. J. Numer. Methods Eng. 39, 2477–2505 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • J.K. Dienes, On the analysis of rotation and stress rate in deforming bodies. Acta Mech. 32, 217–232 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  • P. Dłużewski, Continuum Theory of Dislocations as a Theory of Constitutive Modelling of Finite Elastic-Plastic Deformations. Volume 13 of IFTR Reports. Institute of Fundamental Technological Research – Polish Academy of Science, 1996. (D.Sc. Thesis – in Polish)

    Google Scholar 

  • W. Dornowski, Influence of finite deformations on the growth mechanism of microvoids contained in structural metals. Arch. Mech. 51(1), 71–86 (1999)

    MATH  Google Scholar 

  • W. Dornowski, P. Perzyna, Analysis of the influence of various effects on cycle fatigue damage in dynamic process. Arch. Appl. Mech. 72, 418–438 (2002)

    Article  MATH  Google Scholar 

  • W. Dornowski, P. Perzyna, Numerical investigation of localized fracture phenomena in inelastic solids. Found. Civil Environ. Eng. 7, 79–116 (2006)

    Google Scholar 

  • M.K. Duszek–Perzyna, P. Perzyna, Analysis of the influence of different effects on criteria for adiabatic shear band localization in inelastic solids, vol. 50. Material Instabilities: Theory and Applications (ASME, New York, 1994)

    Google Scholar 

  • A.C. Eringen, Linear theory of nonlocal elasticity and dispersion of plane-waves. Int. J. Eng. Sci. 10(5), 233–248 (1972a)

    Article  MathSciNet  MATH  Google Scholar 

  • A.C. Eringen, Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972b)

    Article  MathSciNet  MATH  Google Scholar 

  • A.C. Eringen, On differential-equations of nonlocal elasticity and solutions of screw dislocation and surface-waves. J. Appl. Phys. 54(9), 4703–4710 (1983)

    Article  Google Scholar 

  • N.A. Fleck, J.W. Hutchinson, Strain gradient plasticity. Adv. Appl. Mech. 33, 295–361 (1997)

    Article  MATH  Google Scholar 

  • M. Frewer, More clarity on the concept of material frame-indifference in classical continuum mechanics. Acta Mech. 202(1–4), 213–246 (2009)

    Article  MATH  Google Scholar 

  • A. Glema, Analysis of Wave Nature in Plastic Strain Localization in Solids. Volume 379 of Rozprawy, Publishing House of Poznan University of Technology, 2004 (in Polish)

    Google Scholar 

  • A. Glema, T. Łodygowski, On importance of imperfections in plastic strain localization problems in materials under impact loading. Arch. Mech. 54(5–6), 411–423 (2002)

    MATH  Google Scholar 

  • A. Glema, W. Kakol, T. Łodygowski, Numerical modelling of adiabatic shear band formation in a twisting test. Eng. Trans. 45(3–4), 419–431 (1997)

    Google Scholar 

  • A. Glema, T. Łodygowski, P. Perzyna, Interaction of deformation waves and localization phenomena in inelastic solids. Comput. Methods Appl. Mech. Eng. 183, 123–140 (2000)

    Article  MATH  Google Scholar 

  • A. Glema, T. Łodygowski, P. Perzyna, Localization of plastic deformations as a result of wave interaction. Comput. Assist. Mech. Eng. Sci. 10(1), 81–91 (2003)

    MATH  Google Scholar 

  • A. Glema, T. Łodygowski, W. Sumelka, P. Perzyna, The numerical analysis of the intrinsic anisotropic microdamage evolution in elasto-viscoplastic solids. Int. J. Damage Mech. 18(3), 205–231 (2009)

    Article  Google Scholar 

  • A. Glema, T. Lodygowski, W. Sumelka, Piotr perzyna – scientific conductor within theory of thermo-viscoplasticity. Eng. Trans. 62(3), 193–219 (2014)

    Google Scholar 

  • A.E. Green, R.S. Rivlin, Multipolar continuum mechanics. Arch. Ration. Mech. Anal. 17(2), 113–147 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  • S. Hanim, J.R. Klepaczko, Numerical study of spalling in an aluminum alloy 7020 – T6. Int. J. Impact Eng. 22, 649–673 (1999)

    Article  Google Scholar 

  • O.M. Heeres, A.S.J. Suiker, R. de Borst, A comparison between the perzyna viscoplastic model and the consistency viscoplastic model. Eur. J. Mech. A. Solids 21(1), 1–12 (2002)

    Article  MATH  Google Scholar 

  • R. Hill, Aspects of invariance in solid mechanics. Adv. Appl. Mech. 18, 1–75 (1978)

    MathSciNet  MATH  Google Scholar 

  • G.A. Holzapfel, Nonlinear Solid Mechanics – A Continuum Approach for Engineering (Chichester, England, 2000)

    MATH  Google Scholar 

  • I.R. Ionescu, M. Sofonea, Functional and Numerical Methods in Viscoplasticity (Oxford University Press, Oxford/New York/Tokyo, 1993)

    MATH  Google Scholar 

  • A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)

    MATH  Google Scholar 

  • J.R. Klepaczko, Dynamic crack initiation, some experimental methods and modelling, in Crack Dynamics in Metallic Materials, ed. by J.R. Klepaczko (Springer, Vienna, 1990), pp. 255–453

    Chapter  Google Scholar 

  • E. Kröner, On the physical reality of torque stresses in continuum mechanics. Int. J. Eng. Sci. 1, 261–278 (1963)

    Article  Google Scholar 

  • Th. Lehmann, Anisotrope plastische Formänderungen. Romanian J. Tech. Sci. Appl. Mech. 17, 1077–1086 (1972)

    MATH  Google Scholar 

  • T. Łodygowski, On avoiding of spurious mesh sensitivity in numerical analysis of plastic strain localization. Comput. Assist. Mech. Eng. Sci. 2, 231–248 (1995)

    Google Scholar 

  • T. Łodygowski, Theoretical and Numerical Aspects of Plastic Strain Localization. Volume 312 of D.Sc. Thesis, Publishing House of Poznan University of Technology, 1996

    Google Scholar 

  • T. Łodygowski, P. Perzyna, Localized fracture of inelastic polycrystalline solids under dynamic loading process. Int. J. Damage Mech. 6, 364–407 (1997a)

    Article  MATH  Google Scholar 

  • T. Łodygowski, P. Perzyna, Numerical modelling of localized fracture of inelastic solids in dynamic loading process. Int. J. Numer. Methods Eng. 40, 4137–4158 (1997b)

    Article  MATH  Google Scholar 

  • T. Łodygowski, W. Sumelka, Anisotropic damage for extreme dynamics, in Handbook of Damage Mechanics Nano to Macro Scale for Materials and Structures, ed. by G.Z. Voyiadjis (Springer, New York, 2015), pp. 1185–1220

    Google Scholar 

  • T. Łodygowski, P. Perzyna, M. Lengnick, E. Stein, Viscoplastic numerical analysis of dynamic plastic shear localization for a ductile material. Arch. Mech. 46(4), 541–557 (1994)

    MATH  Google Scholar 

  • J.K. Mackenzie, The elastic constants of a solids containing spherical holes. Proc. Phys. Soc. 63B, 2–11 (1950)

    Article  MATH  Google Scholar 

  • J.E. Marsden, T.J.H. Hughes, Mathematical Foundations of Elasticity (Prentice-Hall, New Jersey, 1983). https://www.sciencedirect.com/science/article/pii/0079642583900038

    MATH  Google Scholar 

  • M.A. Meyers, C.T. Aimone, Dynamic fracture (Spalling) of materials, Progress in Material Science, 28(1), 1–96 (1983)

    Article  Google Scholar 

  • R.D. Mindlin, Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  • R.D. Mindlin, Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1(4), 417–438 (1965)

    Article  Google Scholar 

  • R.D. Mindlin, N. Eshel, On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4(1), 109–124 (1968)

    Article  MATH  Google Scholar 

  • R.D. Mindlin, H.F. Tiersten, Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11(5), 415–448 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  • W. Moćko, Z.L. Kowalewski, Mechanical properties of a359/sicp metal matrix composites at wide range of strain rates. Appl. Mech. Mater. 82, 166–171 (2011)

    Article  Google Scholar 

  • W. Moćko, Z.L. Kowalewski, Perforation test as an accuracy evaluation tool for a constitutive model of austenitic steel. Arch. Metall. Mater. 58(4), 1105–1110 (2013)

    Article  Google Scholar 

  • J.V. Morán, Continuum Models for the Dynamic Behavior of 1D Nonlinear Structured Solids. Doctoral Thesis, Publishing House of the Universidad Carlos III de Madrid, 2016

    Google Scholar 

  • T. Mura, Micromechanics of Defects in Solids (Kluwer Academic, Dordrecht, 1987)

    Book  MATH  Google Scholar 

  • J.C. Nagtegaal, J.E. de Jong, Some aspects of non-isotropic work-hardening in finite strain plasticity, in Proceedings of the Workshop on Plasticity of Metals at Finite Strain: Theory, Experiment and Computation, ed. by E.H. Lee, R.L. Mallet (Stanford University, 1982), pp. 65–102

    Google Scholar 

  • S. Nemat-Nasser, W.-G. Guo, Thermomechanical response of HSLA-65 steel plates: experiments and modeling. Mech. Mater. 37, 379–405 (2005)

    Article  Google Scholar 

  • J.A. Nemes, J. Eftis, Several features of a viscoplastic study of plate-impact spallation with multidimensional strain. Comput. Struct. 38(3), 317–328 (1991)

    Article  MATH  Google Scholar 

  • J.A. Nemes, J. Eftis, Constitutive modelling of the dynamic fracture of smooth tensile bars. Int. J. Plast. 9(2), 243–270 (1993)

    Article  MATH  Google Scholar 

  • J. Ostrowska-Maciejewska, Mechanika ciał odkształcalnych (PWN, Warszawa, 1994)

    Google Scholar 

  • S.K. Park, X.-L. Gao, Variational formulation of a modified couple stress theory and its application to a simple shear problem. Zeitschrift fur angewandte Mathematik und Physik 59, 904–917 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • P. Perzyna, The constitutive equations for rate sensitive plastic materials. Q. Appl. Math. 20, 321–332 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  • P. Perzyna, Fundamental problems in viscoplasticity. Adv. Appl. Mech. 9, 243–377 (1966)

    Article  Google Scholar 

  • P. Perzyna, Termodynamika materiałów niespreżystych (PWN, Warszawa, 1978) (in Polish)

    Google Scholar 

  • P. Perzyna, Internal state variable description of dynamic fracture of ductile solids. Int. J. Solids Struct. 22, 797–818 (1986a)

    Article  Google Scholar 

  • P. Perzyna, Constitutive modelling for brittle dynamic fracture in dissipative solids. Arch. Mech. 38, 725–738 (1986b)

    MathSciNet  MATH  Google Scholar 

  • P. Perzyna, Instability phenomena and adiabatic shear band localization in thermoplastic flow process. Acta Mech. 106, 173–205 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • P. Perzyna, Constitutive modelling of dissipative solids for localization and fracture, in Localization and Fracture Phenomena in Inelastic Solids, Chapter 3. CISM Course and Lectures, vol. 386, ed. by P. Perzyna (Springer, 1998), pp. 99–241

    Google Scholar 

  • P. Perzyna, The thermodynamical theory of elasto-viscoplasticity. Eng. Trans. 53, 235–316 (2005)

    MathSciNet  MATH  Google Scholar 

  • P. Perzyna, The thermodynamical theory of elasto-viscoplasticity accounting for microshear banding and induced anisotropy effects. Mechanics 27(1), 25–42 (2008)

    Google Scholar 

  • P. Perzyna, The thermodynamical theory of elasto-viscoplasticity for description of nanocrystalline metals. Eng. Trans. 58(1–2), 15–74 (2010)

    Google Scholar 

  • P. Perzyna, Multiscale constitutive modelling of the influence of anisotropy effects on fracture phenomena in inelastic solids. Eng. Trans. 60(3), 225–284 (2012)

    MathSciNet  Google Scholar 

  • I. Podlubny, Fractional differential equations, in Mathematics in Science and Engineering, vol. 198 (Academin Press, USA, 1999)

    MATH  Google Scholar 

  • D. Polyzos, D.I. Fotiadis, Derivation of Mindlin’s first and second strain gradient elastic theory via simple lattice and continuum models. Int. J. Solids Struct. 49, 470–480 (2012)

    Article  Google Scholar 

  • C. Rymarz, Mechanika ośrodków (PWN, Warszawa, 1993) (in Polish)

    Google Scholar 

  • L. Seaman, D.R. Curran, D.A. Shockey, Computational models for ductile and brittle fracture. J. Appl. Phys. 47(11), 4814–4826 (1976)

    Article  Google Scholar 

  • S. Shima, M. Oyane, Plasticity for porous solids. Int. J. Mech. Sci. 18, 285–291 (1976)

    Article  Google Scholar 

  • D.A. Skolnik, H.T. Liu, H.C. Wu, L.Z. Sun, Anisotropic elastoplastic and damage behavior of sicp/al composite sheets. Int. J. Damage Mech. 17, 247–272 (2008)

    Article  Google Scholar 

  • L.J. Sluys, Wave Propagation, Localization and Dispersion in Softening Solids. Doctoral Thesis, Delft University Press, Delft, 1992

    Google Scholar 

  • J.-H. Song, H. Wang, T. Belytschko, A comparative study on finite element methods for dynamic fracture. Comput. Mech. 42, 239–250 (2008)

    Article  MATH  Google Scholar 

  • W. Sumelka, The Constitutive Model of the Anisotropy Evolution for Metals with Microstructural Defects, Publishing House of Poznan University of Technology, Poznań, 2009

    Google Scholar 

  • W. Sumelka, Role of covariance in continuum damage mechanics. ASCE J. Eng. Mech. 139(11), 1610–1620 (2013)

    Article  Google Scholar 

  • W. Sumelka, Fractional viscoplasticity. Mech. Res. Commun. 56,31–36 (2014)

    Article  MATH  Google Scholar 

  • W. Sumelka, A. Glema, The evolution of microvoids in elastic solids, in 17th International Conference on Computer Methods in Mechanics CMM-2007, Łódź-Spała, 19–22 June 2007, pp. 347–348

    Google Scholar 

  • W. Sumelka, A. Glema, Intrinsic microstructure anisotropy in elastic solids, in GAMM 2008 79th Annual Meeting of the International Association of Applied Mathematics and Mechanics, Bremen, 31 Mar–4 Apr 2008

    Google Scholar 

  • W. Sumelka, T. Łodygowski, The influence of the initial microdamage anisotropy on macrodamage mode during extremely fast thermomechanical processes. Arch. Appl. Mech. 81(12), 1973–1992 (2011)

    Article  MATH  Google Scholar 

  • W. Sumelka, T. Łodygowski, Reduction of the number of material parameters by ANN approximation. Comput. Mech. 52, 287–300 (2013)

    Article  Google Scholar 

  • W. Sumelka, M. Nowak, Non-normality and induced plastic anisotropy under fractional plastic flow rule: a numerical study. Int. J. Numer. Anal. Methods Geomech. 40, 651–675 (2016)

    Article  Google Scholar 

  • W. Sumelka, M. Nowak, On a general numerical scheme for the fractional plastic flow rule. Mech. Mater. (2017). https://doi.org/10.1016/j.mechmat.2017.02.005

    Google Scholar 

  • Y. Sun, Y. Shen, Constitutive model of granular soils using fractional-order plastic-flow rule. Int. J. Geomech. 17(8), 04017025 (2017)

    Article  Google Scholar 

  • V.E. Tarasov, General lattice model of gradient elasticity. Mod. Phys. Lett. B 28(17), 1450054 (2014)

    Article  Google Scholar 

  • C. Teodosiu, F. Sidoroff, A theory of finite elastoviscoplasticity of single crystals. Int. J. Eng. Sci. 14(2), 165–176 (1976)

    Article  MATH  Google Scholar 

  • R.A. Toupin, Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11(5), 385–414 (1963)

    MathSciNet  MATH  Google Scholar 

  • R.A. Toupin, Theories of elasticity with couple-stress. Arch. Ration. Mech. Anal. 17(2), 85–112 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  • C. Truesdell, W. Noll, The non-linear field theories of mechanics, in Handbuch der Physik, vol. III/3, ed. by S. Flügge (Springer, Berlin, 1965)

    Chapter  Google Scholar 

  • G.Z. Voyiadjis, F.H. Abed, Microstructural based models for bcc and fcc metals with temperature and strain rate dependency. Mech. Mater. 37, 355–378 (2005)

    Article  Google Scholar 

  • G.Z. Voyiadjis, F.H. Abed, Implicit algorithm for finite deformation hypoelastic-viscoplasticity in fcc metals. Int. J. Numer. Methods Eng. 67, 933–959 (2006)

    Article  MATH  Google Scholar 

  • G.Z. Voyiadjis, R.K. Abu Al-Rub, Gradient plasticity theory with a variable length scale parameter. Int. J. Solids Struct. 42(14), 3998–4029 (2005)

    Article  MATH  Google Scholar 

  • G.Z. Voyiadjis, P.I. Kattan, Evolution of fabric tensors in damage mechanics of solids with micro-cracks: part I – theory and fundamental concepts. Mech. Res. Commun. 34, 145–154 (2007)

    Article  MATH  Google Scholar 

  • H. Xiao, O.T. Bruhns, A. Meyers, Hypo-elasticity model based upon the logarithmic stress rate. J. Elast. 47, 51–68 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • H. Xiao, O.T. Bruhns, A. Meyers, Strain rates and material spin. J. Elast. 52, 1–41 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • R. Xiao, H. Sun, W. Chen, A finite deformation fractional viscoplastic model for the glass transition behavior of amorphous polymers. Int. J. Non Linear Mech. 93, 7–14 (2017)

    Article  Google Scholar 

  • F. Yang, A.C.M. Chong, D.C.C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)

    Article  MATH  Google Scholar 

  • S. Zaremba, Sur une forme perfectionée de la théorie de la relaxation. Bull. Int. Acad. Sci. Cracovie 594–614 (1903)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Structural Engineering, Poznan University of Technology, Poznan, Poland

    Wojciech Sumelka & Tomasz Łodygowski

Authors
  1. Wojciech Sumelka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tomasz Łodygowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wojciech Sumelka .

Editor information

Editors and Affiliations

  1. Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA, USA

    George Z. Voyiadjis

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Sumelka, W., Łodygowski, T. (2019). Implicit Nonlocality in the Framework of Viscoplasticity. In: Voyiadjis, G. (eds) Handbook of Nonlocal Continuum Mechanics for Materials and Structures. Springer, Cham. https://doi.org/10.1007/978-3-319-58729-5_17

Download citation

Publish with us

Policies and ethics

Search

Navigation