Skip to main content
Log in

On nonlocal regularization in one dimensional finite strain elasticity and plasticity

  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

The purpose of this article is to explore in details the theoretical and numerical aspects of the behavior of spatial trusses, undergoing large elastic and/or elastoplastic strains. Two nonlocal formulations are proposed in order to regularize the problem, avoiding the mesh dependence of the numerical response. The classical example of a simple bar in tension is chosen to explore the various features of the proposed models and to highlight the interplay between material and geometrical nonlinearity in the localization.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Nadai A (1931) Plasticity McGraw-Hill New York

  • Asaro R J (1985) Material modeling and failure modes in metal plasticity mechanics of materials. Mech Materials 4:343–373

    Article  Google Scholar 

  • Olmstead W E, Nemat-Nasser S, Ni L (1994) Shear bands as surfaces of discontinuity. J Mech Phys Solids 42(4):697–709

    Article  CAS  Google Scholar 

  • McMeeking R M, Rice J R (1975) Finite element formulations for problems of large elasto-plastic deformation. Int J Solids Struct 11:601–616

    Article  Google Scholar 

  • Benallal A, Billardon R, Geymonat G (1988) Some mathematical aspects of the damage softening problems. In: Mazars J, Bazant Z P (eds) Cracking and Damage 247–258. Elsevier, Amsterdam

  • Rizzi E, Maier G, Willam K (1996) On failure indicators in multidissipative materials Int J Solids Struct 33(20–22):3187–3214

    Google Scholar 

  • Frémond M, Nedjar B (1996) Damage gradient of damage and principle of virtual power. Int J Solids Struct 33(8)1083–1103

    Google Scholar 

  • de Borst R, Heeres O M, Benallal A (1995) A gradient enhanced damage model: theory and computation Computational In: Owen D.R.J., Onate E, Hinton E (eds.) Plasticity: Fundamentals and applications Complas IV pp 975–986 Pineridge-Press

  • Comi C, Driemeier L (1998) On gradient regularization for numerical analysis in the presence of damage. In: de Borst R. van der Giessen E., (eds) Material Instabilities in Solids - chapter XVI pp 425–440

  • Comi C (1999) Computational modelling of gradient-enhanced damage in quasi-brittle materials. Mech Cohes - Frict Mater 4(1):17–36

    Google Scholar 

  • Bazant Z P (1987) Why continuum damage is nonlocal: micromechanics arguments. J Eng Mech 117(5):1070–1087

    Google Scholar 

  • Pijaudier-Cabot G, Bazant Z L (1987) Nonlocal damage theory. J Eng Mech 113:1512–1533

    Google Scholar 

  • Comi C (2001) A non-local model with tension and compression damage mechanisms. Euro J Mech A/Solids 20:1–22

    Google Scholar 

  • Jirasek M (1998) Nonlocal models for damage and fracture: comparison ofapproaches. Int J Solids Struct 35:4133–4145

    Article  Google Scholar 

  • Steinmann P, Stein E (1994) Finite element localization analysis of micropolar strength degrading materials. In: Mang H, Bicanic N, de Borst R. (eds.) Computer modelling of concrete structures North-Holland Publ Comp 435–444

  • Benallal A, Tvergaard V (1995) Nonlocal continuum effects on bifurcation in the plane strain tension-compression test. J Mech Phys Solids 43(5):741–770

    Article  Google Scholar 

  • Steinmann P (1996) On localization analysis in multisurface hyperelasto-plasticity. J Mech Phys Solids 44(10):1691–1713

    Article  CAS  Google Scholar 

  • Steinmann P (1999) Formulation and computation of geometrically non-linear gradient damage. Int J Numer Meth Eng 46:757–779

    Article  Google Scholar 

  • Brunig M, Ricci S, Obrecht H (2001) Nonlocal large deformation and localization behavior of metals. Compu Struct 79:2063–2074

    Article  Google Scholar 

  • Areias PMA, César de Sá JMA, António CAC (2003) A gradient model for finite strain elastoplasticity coupled with damage. Finite Eleme Anal Design 39:1191–1235

    Article  Google Scholar 

  • Bazant ZP, Belytschko T (1984) Wave propagation in a strain softening bar:exact solution. J Eng Mech 111(3):381–389

    Google Scholar 

  • Ogden RW (1997) Nonlinear elastic deformations Dover Publications Inc.

  • Driemeier L, Proença SPB, Alves M (2004) A contribution to the numerical nonlinear analysis of three–dimensional truss systems considering largestrains damage and plasticity in press. Commun Nonlinear Solids Numer Simul

  • Peric D, Owen DRJ, Honnor ME (1992) A model for finite strain elasto-plasticity based on logarithmic strains: computational issues. Comput Meth Appl Mech Eng 94:35–61

    Article  Google Scholar 

  • Bruhns OT, Xiao H, Meyers A (2001) Constitutive inequalities for an isotropic elastic strain–energy function based on Hencky’s logarithmic strain tensor. Proc. R. Soc. Lond. A, 457:2207–2226

    Google Scholar 

  • Hill R (1958) A general theory of uniqueness and stability of materials in elastic-plastic solids. J Mech Phys Solids 6:236–249

    Article  Google Scholar 

  • Drucker DC (1964) On the postulate of stability of materials in the mechanics of continua. J. de mécanique 3(2):235–249

    Google Scholar 

  • Maier G, Hueckel (1979) Nonassociated and coupled flow rules of elastoplasticity for rock-like Materials. Int J rock mech mining sciences geomech abstract 16:77–92

    Article  Google Scholar 

  • Edelen DGB, Laws N (1971) On the thermodynamics of systems with nonlocality. Arch Ratio Mech Anal 43:36–44

    Google Scholar 

  • Edelen DGB, Green AE, Laws N (1971) On the thermodynamics of systems with nonlocality. Arch Ratio Mech Anal 43:24–35

    Google Scholar 

  • Cemal Eringen A (1972) Linear theory of nonlocal elasticity and dispersion of plane waves. Int J Eng Science 10:425–435

    Article  Google Scholar 

  • Rodríguez-Ferran A, Morata I, Huerta A (2004) Efficient and reliable nonlocal damage models. Comput Meth Appl Mech Eng 193:3431–3455

    Article  Google Scholar 

  • Zienkiewicz OC, Taylor RL (2000) The finite element method 5th edition Vol 2 Butterworth Heinemann

  • Comi C, Perego U (1996) A generalized variable formulation for gradient–dependent softening plasticity. Int J Numer Meth Eng 39:3731–3755

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to L. Driemeier.

Additional information

Aknowledgement The financial support of FAPESP, a Brazilian research funding agency and of the italian miur, project prin 2003082105, are greatly appreciated.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Driemeier, L., Comi, C. & Proença, S. On nonlocal regularization in one dimensional finite strain elasticity and plasticity. Comput Mech 36, 34–44 (2005). https://doi.org/10.1007/s00466-004-0640-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-004-0640-7

Keywords

Navigation