Summary
We presented aspects and results related to the broad field of strain localization with special focus on large strain elastoplastic response. Therefore, we first re-examined issues related to the classification of discontinuities and the classical description of localization with a particular emphasis on an Eulerian geometric representation. We touched the problem of mesh objectivity and discussed results of a particular regularization method, namely the micropolar approach. Generally, regularization has to preserve ellipticity and to reflect the underlying physics. For example ductile materials have to be modelled including viscous effects whereas geomaterials are adequately described by the micropolar approach. Then we considered localization phenomena within solids undergoing large strain elastoplastic deformations. Here, we documented the influence of isotropic damage on the failure analysis. Next, the interesting influence of an orthotropic yield condition on the spatial orientation of localized zones has been studied. Finally, we investigated the localization condition for an algorithmic model of finite strain single crystal plasticity.
Similar content being viewed by others
References
Aifantis, E. C. 1984: On the Microstructural Origin of Certain Inelastic Models, J. Engr. Mat. Tech., ASME, 106: 326–334
Aifantis, E. C. 1987: The Physics of Plastic Deformation. Int. J. Plast., 3: 211–247
Asaro, R. J. 1983: Crystal Plasticity, J. Appl. Mech., ASME, 50: 921–934
Bažant, Z. P.; Belytschko, T. B. 1985: Wave Propagation in a Softening Bar: Exact Solution. J. Engr. Mech., ASCE, 111: 381–389
Bažant, Z. P.; Lin, F. B. 1988: Non-Local Yield Limit Degradation. Int. J. Num. Meth. Engr., 26: 1805–1823
Benallal A., Billardon, R.; Geymonat, G. 1991: Localization Phenomena at the Boundaries and Interfaces of Solids, in Proc. 3rd Int. Conf. on Constitutive Laws for Engineering Materials, Tucson 1991, Eds. C. S. Desai et al. ASME Press, New York
Betsch P., Gruttmann, F.; Stein, E. 1994: A 4-Node Finite Shell Element for the Implementation of General Hyperelastic 3D-Elasticity at Finite Strains. Comp. Meth. Appl. Mech. Engr., submitted
Borst R. De 1991: Simulation of Strain Localization: A Reappraisal of the Cosserat Continuum. Engr. Comp., 8: 317–332
Borst, R.De 1993: A Generalisation of J 2-Flow Theory for Polar Continua. Comp. Meth. Appl. Mech. Engr., 103: 347–362
Borst, R.De; Feenstra, P. H. 1990: Studies in Anisotropic Plasticity with Reference to the Hill Criterion. Int. J. Num. Meth. Eng., 29: 315–336
Cosserat, E.; Cosserat, F. 1909: Théorie des Corps Déformables, Librairie Scientifique A. Hermann et Fils, Paris
Dietsche, A.; Steinmann, P.; Willam, K. 1992: Micropolar Elasto-Plasticity and its Role in Localization Analysis. Int. J. Plast., 9: 813–831
Duszek, M. K.; Perzyna, P.; Stein, E. 1992: Adiabatic Shear Band Localization in Elastic-Plastic Damaged Solids. Int. J. Plast. 8: 361–384
Gurson, A. L. 1975: Plastic Flow and Fracture Behaviour of Ductile Materials Incorporating Void Nucleation, Growth and Interaction, Ph.D. Thesis, Division of Engineering, Brown University, Providence
Gurson, A. L. 1977: Continuum Theory of Ductile Rupture by Void Nucleation and Growth-Yield Criteria and Flow Rules for Porous Ductile Media, J. Engr. Mat. Techn., 99: 2–15
Hadamard, J. 1903: Leçons sur la propagation des ondes et les équation de l'hydrodynamique, Librairie Scientifique A. Hermann et Fils, Paris
Hill, R. 1948: A Theory of the Yielding and Plastic Flow of Anisotropic Materials. Proc. Roy. Soc., A 193: 281–297
Hill, R. 1958: A General Theorie of Uniqueness and Stability in Elastic-Plastic Solids. J. Mech. Phys. Solids, 6: 236–249
Hill, R. 1962: Acceleration Waves in Solids. J. Mech. Phys. Solids, 10: 1–16
Larsson, R.; Runesson, K.; Ottosen, N. S. 1993: Discontinous Displacement Approximation for Capturing Plastic Localization. Int. J. Num. Meth. Engr., 36: 2087–2105
Lee, E. H. 1969: Elastic-Plastic Deformations at Finite Strains. J. Appl. Mech., ASME 36: 1–6
Lemaitre, J. 1984: How to Use Damage Mechanics. Nucl. Engr. and Design, 80: 233–245
Lemaitre, J. 1985: Coupled Elasto-Plasticity and Damage Constitutive Equations. Comp. Meth. Appl. Mech. Engr., 51: 31–49
Lemaitre, J. 1985: A Continuous Damage Mechanics Model for Ductile Fracture. J. Engr. Mat. Techn., 107: 83–89
Levitas V. I.; Stein, E. 1994: Thermodynamical Model of Martensitic Phase Transitions. ZAMM, accepted
Levitas V. I.; Stein, E.; Lengnick, M. 1994: A Unified Approach for the Description of Strain Localization and Phase Transitions. Int. J. Plasticity, submitted
Loret, B.; Prevost, J. H. 1990: Dynamic Strain Localization in Elasto-(Visco) Plastic Solids, Part I. General Formulation and One-Dimensional Examples. Comp. Meth. Appl. Mech. Eng., 83: 247–273
Mandel, J. 1962: Ondes plastiques dans un milieu indéfini à trois dimensions. J. Mécanique, 1: 3–30
Mandel, J. 1966: Conditions de Stabilité et Postulat de Drucker. In Rheology and Soil Mechanics, Eds. J.Kravtchenko and P. M.Sirieys, Springer, Berlin etc
Miehe, C. 1993: Computation of Isotropic Tensor Functions. Comm. Num. Meth. Engr., 9: 889–896
Miehe, C. 1994: On the Representation of Prandtl-Reuss-Tensors within Multiplicative Elastoplasticity. Int. J. Plast., 40: 609–621
Miehe, C. 1992: Aspects of the Formulation and Numerical Implementation of Large Strain Isotropic Elasticity. Int. J. Num. Meth. Engr., 37: 1981–2004
Miehe, C.; Stein, E. 1992: A Canonical Model of Multiplicative Elasto-Plasticity. Formulation and Aspects of the Numerical Implementation. Eur. J. Mech. A/Solids, 11: 25–43
Miehe, C.; Stein, E.; Wagner, W. 1994: Associative Multiplicative Elastoplasticity. Formulation and Aspects of the Numerical Implementation Including Stability Analysis. Comp. Struct., 52: 969–978
Miehe, C.; Schröder, J. 1994: Post-Critical Discontinuous Localization Analysis of Small-Strain Softening Elastoplastic Solids. Arch. Appl. Mech., 64: 267–285
Mühlhaus, H. B.; 1989: Application of Cosserat Theory in Numerical Solutions of Limit Load Problems. Ing. Arch., 59: 124–137
Mühlhaus, H. B.; Aifantis, E. C. 1991: A Variational Principle for Gradient Plasticity. Int. J. Solids Struct., 28: 845–858
Mühlhaus, H. B.; Vardoulakis, I. 1987: The Thickness of Shear Bands in Granular Materials. Géotechnique, 37: 271–283
Needleman, A. 1988: Material Rate Dependence and Mesh Sensitivity in Localization Problems. Comp. Meth. Appl. Mech. Eng., 67: 69–86
Ortiz, M.; Leroy, Y.; Needlman, A. 1987: A Finite Element Method for Localization Failure Analysis. Comp. Meth. Appl. Mech. Engr., 61: 189–214
Perzyna, P. 1966: Fundamental Problems in Viscoplasticity. Adv. Appl. Mech., 9: 243–377
Pietruszak, S.; Mroz, Z. 1981: Finite Element Analysis of Deformation of Strain Softening Materials. Int. J. Num. Meth. Engr., 17: 327–334
Pijaudier-Cabot, G.; Bažant, Z. P. 1987: Nonlocal Damage Theory. J. Engr. Mech., ASCE, 113: 1512–1533
Pijaudier-Cabot, G.; Benallal, A. 1993: Strain Localization and Bifurcation in a Nonlocal Continuum. Int. J. Solids Struct., 30: 1761–1775
Rice, J. R. 1976: The Localization of Plastic Deformation. In Theoretical and Applied Mechanics, Ed. W. T.Koiter, North Holland, Amsterdam etc
Rudnicki, J. W.; Rice, J. R. 1975: Conditions for the Localization of Deformation in Pressure-Sensitive Dilatant Materials. J. Mech. Phys. Solids, 23: 371–394
Simo, J. C. 1992: Algorithms for Static and Dynamic Multiplicative Plasticity that Preserve the Classical Return Mapping Schemes of the Infinitesimal Theory. Comp. Meth. Appl. Mech. Engr., 99: 61–112
Simo, J. C.; Armero, F. 1992: Geometrically Nonlinear Enhanced Strain Mixed Method and the Method of Incompatible Modes. Int. J. Num. Meth. Engr., 33: 1413–1449
Simo, J. C.; Armero, F.; Taylor, R. L. 1993: Improved Versions of Assumed Enhanced Strain Tri-Linear Elements for 3D Finite Deformation Problems. Comp. Meth. Appl. Mech. Engr., 110: 359–386
Simo, J. C.; Miehe, C. 1992: Associated Coupled Thermoplasticity at Finite Strains: Formulation, Numerical Analysis and Implementation. Comp. Meth. Appl. Mech. Engr., 98: 41–104
Simo, J. C.; Oliver, J.; Armero, F. 1993: An Analysis of Strong Discontinuities Induced by Strain-Softening in Rate-Independent Inelastic Solids. Comp. Mech., 12: 277–296
Simo, J. C.; Rifai, M. S. 1990: A Class of Mixed Assumed Strain Methods and the Method of Incompatible Modes. Int. J. Num. Meth. Engr., 29: 1595–1638
Sluys, L. J.; DeBorst, R. 1992: Wave Propagation and Localisation in a Rate-Dependent Cracked Medium: Model Formulation and One-Dimensional Examples. Int. J. Solids Struct., 29: 2945–2958
Steinmann, P.; Willam, K. 1991: Performance of Enhanced Finite Element Formulations in Localized Failure Computations. Comp. Meth. Appl. Mech. Engr., 90: 845–867
Steinmann, P.; Willam, K. 1991: Localization within the Framework of Micropolar Elasto-Plasticity, in Advances in Continuum Mechanics, Eds. O.Brüller, V.Mannl and J.Najar, Springer, Berlin etc.
Steinmann, P.; Stein, E. 1994: Finite Element Localization Analysis of Micropolar Strength Degrading Materials. Proc. EUROC'94, Innsbruck, Austria, Eds. H. Mang, N. Bicanic and R. de Borst
Steinmann, P.; Miehe, C.; Stein, E. 1993: Comparison of Different Finite Deformation Inelastic Damage Models within Multiplicative Elastoplasticity for Ductile Materials. Comp. Mech., 13 458–474
Steinmann, P.; Miehe, C.; Stein, E. 1994: On the Localization Analysis of Orthotropic Hill Type Elastoplastic Solids. J. Phys. Mech. Solids, 42: 1969–1994
Steinmann, P.; 1995: An Improved FE Expansion for Micropolar Localization Analysis. Comm. Num. Meth. Engr., in press
Steinmann, P.; Stein, E. 1995: On the Numerical Treatment and Analysis of Finite Deformation Ductile Single Crystal Plasticity. Comp. Meth. Appl. Mech. Engr., submitted
Steinmann, P.; Betsch, P.; Stein, E. 1995: FE Plane Stress Analysis Incorporating Arbitrary 3D Large Strain Constitutive Models. Engr. Comp., submitted
Steinmann, P.; Stein, E. 1995: Numerical Analysis of Localization Phenomena in Single Crystal Plasticity. Proc. Complas IV '95, Barcelona, Spain
Thomas, T. Y. 1961: Plastic Flow and Fracture of Solids, Academic Press, New York etc.
Vardoulakis, I.; Aifantis, E. C. 1989: Gradient Dependent Dilatancy and its Implications in Shear Banding and Liquefaction. Ing. Arch., 59: 197–208
Willam, K.; Montgomery, K. 1987: Fracture Energy-Based Softening Plasticity Model for Shear Failure. Proc. Int. Symp. on Interaction of Conventional Munition with Protective Structures, Mannheim 1987, 2: 679–691
Author information
Authors and Affiliations
Additional information
Communicated by S. N. Atluri, 18 August 1995
Dedicated to the Memory of Juan Carlos Simo.
Extended Version of an Invited Lecture at the 4th International Conference on Computational Plasticity COMPLAS IV, April 3.–6. '95, Barcelona, Spain
Rights and permissions
About this article
Cite this article
Stein, E., Steinmann, P. & Miehe, C. Instability phenomena in plasticity: Modelling and computation. Computational Mechanics 17, 74–87 (1995). https://doi.org/10.1007/BF00356480
Issue Date:
DOI: https://doi.org/10.1007/BF00356480