Numerical Solutions of Initial Boundary Value Problems for Metals and Soils

  • T. Lodygowski
Part of the International Centre for Mechanical Sciences book series (CISM, volume 386)


This work considers the numerical aspects of the problems of plastic strain localization for strain softening materials in both soil-like materials and ductile metals. In mathematical formulation the attention should be paid on the well-posedness of the initial boundary value problem which guarantees the uniqueness of the solution in the whole domain of the incremental analysis. Viscoplasticity serves as a mathematical tool of regularization of the system of governing equations. This formulation which is the base of numerics allows to stay free of the effects of the results sensitivity to the finite element mesh density. The numerical examples for the initial boundary value problems of ductile and clay-type materials are presented.

There are also presented some preliminary thoughts on the sensitivity analysis for these ruther complicated problems of plastic strain localization in softening materials.


Initial Boundary Plastic Equivalent Strain Cauchy Stress Tensor Length Scale Parameter Initial Boundary Value Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    ABAQUS. Manuals for v. 5.6. Reports, Hibbitt, Karlsson Sorensen, Inc., 1997.Google Scholar
  2. [2]
    T. ADACHI and F. OKA. Constitutive equations for normally consolidated clay based on elasto-viscoplasticity. Solids and Foundations, 22 (4): 57–70, 1982.CrossRefGoogle Scholar
  3. [3]
    T. ADACHI, F. OKA, and M. MIMURA. Mathematical structure of an overstress elasto-viscoplastic model for clay. Soils and Foundations, 27 (4): 31–42, 1987.CrossRefGoogle Scholar
  4. [4]
    R. C. BATRA. Numerical solutions of initial-boundary-value problems with shear band localization. Proceedinngs of Advanced School on Localization and fracture phenomena in inelastic solids, Udine (in this volume), 1997.Google Scholar
  5. [5]
    R. C. BATRA and X. ZHANG. On the propagation of shear band in a steel tube. Int.Jour. of Plasticity, 1993.Google Scholar
  6. [6]
    Z. P. BAZANT, J. PAN, and G. PIJAUDIER-CABOT. Softening in reinforced concrete beams and frames.Google Scholar
  7. [7]
    T. BELYTSCHKO and J. FISH. Spectral superposition on finite elements for shear banding problems. In V Int. Symp. on Numerical Methods in Engineering, Lausanne, 11–15 Sept. 1989, Part I. Google Scholar
  8. [8]
    T. BELYTSCHKO and J. FISH. Embedded hinge lines for plate elements. Comp. Meth. in Appl. Mech. Eng.,76:67–86, 1989.Google Scholar
  9. [9]
    T. BELYTSCHKO, J. FISH, and B. E. ENGELMANN. A finite element with embedded localization zones. Comp. Meth. in Appl. Mech. Eng.,70:59–89, September 1988.Google Scholar
  10. [10]
    T. BELYTSCHKO and D. LASRY. A study of localization limiters for strain-softening in statics and in dynamics. Comp. Struc., 33 (3): 707–715, 1989.CrossRefMATHGoogle Scholar
  11. [11]
    T. BELYTSCHKO, X.-J. WANG, Z. P. BAZANT, and Y. HYUN. Transient solutions for one-dimensional problems with strain softening. Jour. Appl. Mech., 54: 513–518, September 1987.CrossRefMATHGoogle Scholar
  12. [12]
    D. BIGONI and T. HUECKEL. Uniqueness and localization—I. associative and non-associative elastoplasticity. Int. Jour. of Plasticity, 28 (2): 197–213, 1991.MathSciNetMATHGoogle Scholar
  13. [13]
    R. de BORST. Simulation of strain localization: A reappraisal of the Cosserat continuum. Eng. Comp., 8: 317–332, 1991.CrossRefGoogle Scholar
  14. [14]
    R. de BORST. Gradient-dependent plasticity: Formulation and algorithmic aspects. Int. J. Num. Meth. in Eng.,35:521–539, 1992.Google Scholar
  15. [15]
    R. de BORST. Fundamental issues in finite element analyses of localization of deformation. Eng. Comp., 10: 99–121, 1993.CrossRefGoogle Scholar
  16. [16]
    R. de BORST, H.-B. MUEHLHAUS, J. PAMIN, and L. J. SLUYS. Computational modelling of localisation of deformation. In D. R. J. Owen, E. Onate, and E. Hinton, editors, Proc. Third Intern. Conference on Computational Plasticity, Fundamentals and Applications, Barcelona, 1992,pages 483–508, Swansea, April 4–9 1992. Pineridge Press.Google Scholar
  17. [17]
    A. K. CHAKRABARTI and J. W. SPRETNAK. Instability of plastic flow in the directions of pure shear: I. theory. Metallurgica Transactions, 6A: 733–747, April 1975.Google Scholar
  18. [18]
    D. R. CURRAN, L. SEAMAN, and D.A. SHOCKEY. Dynamic failure in solids. Physics Today, pages 46–55, 1977.Google Scholar
  19. [19]
    J. DESRUES. La localisation de la deformation dans les materiaux granulaires. PhD thesis, L’Universite Scientifique et Medical et L’Institut National Polytechnique de Grenoble, 1984.Google Scholar
  20. [20]
    J. DESRUES and R. CHAMBON. Shear band analysis for granular materials: The question of incremental non-linearity. Ingenieur-Archiv, 59: 187–196, 1989.CrossRefGoogle Scholar
  21. [21]
    A. DRESCHER. Zagadnienia doswiadczalnej weryfikacji modelu ciala o wzmocnieniu gestosciowym. Rozp. Inz., 3: 351–387, 1972.Google Scholar
  22. [22]
    M. DUSZEK-PERZYNA, A. GARSTECKI, A. GLEMA, and T. LODYGOWSKI. Sensitivity of strain localization–parametric study. In A.Gastecki and J.Rakowski, editors, Proceedings of Polish Conference on Computer Methods in Mechanics, PCCMM’97–Poznan, volume 1, pages 345–352, 1997.Google Scholar
  23. [23]
    G. DUVAUT and J. L. LIONS. Inequalities in Mechanics and Physics. Springer, Berlin [u.a.], 1976.CrossRefMATHGoogle Scholar
  24. [24]
    G. ENGELN-MÜLLGES and F. REUTER. Formelsammlung zur Numerischen Mathematik mit Standard-FORTRAN 77-Programmen. BI Wissenschaftsverlag, 1988.Google Scholar
  25. [25]
    J. FISH and T. BELYTSCHKO. A general finite element procedure for problems with high gradients. Comp. Struc., 35 (4): 309–319, 1990.CrossRefMATHGoogle Scholar
  26. [26]
    A. GARSTECKI and A. GLEMA. Sensitivity analysis and optimal redesign of columns in the state of initial distortions and prestress. Structural Optimization, 3, 1991.Google Scholar
  27. [27]
    A. GAWUCKI. Spr@zysto—plastyczne konstrukcje pretowe z luzami. Rozprawy Politechniki Poznariskiej, (185), 1987.Google Scholar
  28. [28]
    A. GAWCKI and B. JANIIVSKA. Problemy analizy i identyfikacji w procesach nieustalonego przeplywu ciepla. ZNPP, 39: 103–131, 1995.Google Scholar
  29. [29]
    A. GLEMA, W. KAKOL, and T. LODYGOWSKI. Numerical modelling in adiabatic shear band formation in a twisting test. Eng. Trans., 3, 1997.Google Scholar
  30. [30]
    A. GLEMA and T. LODYGOWSKI. Plastic strain localization and failure of a ductile specimen. In ZAMM’96, volume 77, pages S97–98, 1997.Google Scholar
  31. [31]
    A. GLEMA, T. LODYGOWSKI, and P. PERZYNA. Effects of microdamage in plastic strain localization. In A.Gastecki and J.Rakowski, editors, Proceedings of Polish Conference on Computer Methods in Mechanics, PCCMM’97–Poznan, volume 2, pages 451–458, 1997.Google Scholar
  32. [32]
    M. E. GURTIN. An Introduction to Continuum Mechanics. Academic Press, 1981.Google Scholar
  33. [33]
    J. HADAMARD. Lesons sur la Propagation des Ondes et les Equations de L’Hydrodynamique. Paris, 1903.Google Scholar
  34. [34]
    R. HILL. A general theory of uniqueness and stability in elastic—plastic solids. J. Mech. Phys. Solids, 6: 236–249, 1958.CrossRefMATHGoogle Scholar
  35. [35]
    R. HILL. Acceleration waves in solids. J. Mech. Phys. of Solids, 10: 1–16, 1962.CrossRefMATHGoogle Scholar
  36. [36]
    R. HILL and J. W. HUTCHINSON. Bifurcation phenomena in the plane tension test. J. Mech. Phys. Solids, 23, 1975.Google Scholar
  37. [37]
    T. J. R. HUGHES, T. KATO, and J. E. MARSDEN. Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Rat. Mech. Anal., 63: 273–294, 1977.MathSciNetMATHGoogle Scholar
  38. [38]
    IONESCU I.R. and M. SOFONEA. Functional and numerical methods in viscoplasticity. Oxford University Press, Oxford, New York, Tokyo, 1993.Google Scholar
  39. [39]
    T. KATO. The Cauchy problem for quasi—linear symmetric hyperbolic systems. Arch. Rational Mech. Anal., 58: 181–205, 1975.MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    K. KIBLER, M. LENGNICK, and T. LODYGOWSKI. Selected aspects of the well—posedness of the localized plastic flow processes. CAM é9 ES (submitted for publication), 1994.Google Scholar
  41. [41]
    M. KLEIBER. Shape and non-shape structural sensitivity analysis for problems with any material and kinematic non-linearity. Comp. Meth. in Appl. Mech. Eng.,108:73–97, 1993.Google Scholar
  42. [42]
    M. KLISIIVSKI, K. RUNESSON, and S. STURE. Finite element with inner softening band. Jour. of Eng. Mech., ASCE, 17 (3): 575–587, 1991.Google Scholar
  43. [43]
    A. KORBEL. Structural and mechanical aspects of homogeneous and heterogeneous deformation of solids. Proceedinngs of Advanced School on Localization and fracture phenomena in inelastic solids, Udine (this volume), 1997.Google Scholar
  44. [44]
    S. KURCYUSZ. Matematyczne podstawy teorii optymalizacji. PWN - Warszawa, 1982.Google Scholar
  45. [45]
    E.H. LEE. Elastic—plastic deformations at finite strains. J. Appl. Mech., 36, 1969.Google Scholar
  46. [46]
    J. LEMAITRE and J.-L. CHABOCHE. Mechanics of solid materials. Cambridge University Press, 1985.Google Scholar
  47. [47]
    M. LENGNICK, T. LODYGOWSKI, P. PERZYNA, and E. STEIN. On regularization of plastic flow localization in a soil material. Eng. Trans., 44 (3), 1996.Google Scholar
  48. [48]
    S. LEROUEIL, M. KOBBAJ, F. TAVENAS,, and R. BOUCHARD. Stress-strainstrain rate relation for the compressibility of sensitive natural clays. Geotechnique, 35: 159–180, 1985.CrossRefGoogle Scholar
  49. [49]
    T. LODYGOWSKI. Numerical analysis of softening structures. In V Int. Symp. on Numerical Methods in Engineering, Lausanne, 11–15 Sept. 1989, Part I, pages 371–376, 1989.Google Scholar
  50. [50]
    T. LODYGOWSKI. Mesh independent beam elements for strain localization. Comp. Meth. in Civil Eng.,3(3):9–24, 1993.Google Scholar
  51. [51]
    T. LODYGOWSKI. On avoiding of spurious mesh sensitivity in numerical analysis of plastic strain localization. CAM ê9 ES, 2 (3): 231–248, 1995.Google Scholar
  52. [52]
    T. LODYGOWSKI. Theoretical and numerical aspects of plastic strain localization. Wyd. politechniki Poznanskiej, 1996.Google Scholar
  53. [53]
    T. LODYGOWSKI and A. GLEMA. Numerial modelling of plastic strain localization in an adiabatic twisting test. In Cz. Rymarz, editor, XII Conf. CMM, Warsaw-Zegrze, May 9–13, 1995, 1995.Google Scholar
  54. [54]
    T. LODYGOWSKI, M. LENGNICK, P. PERZYNA, and E. STEIN. Viscoplastic numerical analysis of dynamic plastic strain localization for a ductile material. Arch. Mech., 46 (4): 541–557, 1994.MATHGoogle Scholar
  55. [55]
    T. LODYGOWSKI, M. LENGNICK, and E. STEIN. Numerical analysis of localization phenomena in ductile and brittle materials. In B.H.V. Topping, editor, Second International Conference on Computational Structures Technology, Athena, Greece, August 30 - September 1, 1994, 1994.Google Scholar
  56. [56]
    T. LODYGOWSKI, M. LENGNICK, and E. STEIN. On rate dependent regularization of localization phenomena in a brittle-degradating material. In R. de Borst N. Bicanic, H. Mang, editor, EURO-C’94 Conference on Numerical Modelling of Concrete Structures, Innsbruck, March 22–25, 1994, volume 1, pages 333–342, 1994.Google Scholar
  57. [57]
    T. LODYGOWSKI and P. PERZYNA. Localized fracture in inelastic polycrystalline solids under dynamic loading processes. Int. Journ. Damage Mech., 6 (4): 364, 1997.CrossRefGoogle Scholar
  58. [58]
    T. LODYGOWSKI and P. PERZYNA. Numerical modelling of localized fracture of inelastic solids in dynamic loading processes. Int. Journ. Num. Meth. Eng. Sci., 40, 1997.Google Scholar
  59. [59]
    B. LORET. An introduction to classical theory of elastoplasticity. In F.Darve, editor, Geomaterials: constitutive equations and modelling, pages 149–186. Elsevier Applied Science, London and New York, 1990.Google Scholar
  60. [60]
    A.M. LYAPUNOV. The general problem of the stability of motion. Int. Journ. Control, 55, 1992.Google Scholar
  61. [61]
    J. MANDEL. Conditions de stabilite et postulat de Drucker. In J. Kravtchenko and P.M. Sirieys, editors, Rheology and soil mechanics, pages 58–68. Springer, Berlin, 1966.Google Scholar
  62. [62]
    A. MARCHAND and J. DUFFY. An experimental study of the formation process of adiabatic shear bands in a structural steel. J. Mech. Phys. of Solids, 1987.Google Scholar
  63. [63]
    J. C. NAGTEGAAL. On the implementation of inelastic constitutive equations with special reference to large deformation problems. Comp. Meth. Appl. Mech. Eng., 33: 469–486, 1982.CrossRefMATHGoogle Scholar
  64. [64]
    A. NEEDLEMAN. Material rate dependence and mesh sensitivity in localization problems. Comp. Meth. in Appl. Mech. Eng.,67:69–85, 1988.Google Scholar
  65. [65]
    M. K. NEILSEN and H. L. SCHREYER. Bifurcation in elastic-plastic materials. Int. J. Solids Struc., 30 (4): 521–544, 1993.CrossRefMATHGoogle Scholar
  66. [66]
    M. ORTIZ, Y. LEROY, and A. NEEDLEMAN. A finite element method for localized failure analysis. Comp. Meth. in Appl. Mech. Eng.,61, 1987.Google Scholar
  67. [67]
    H. OUMERACI. Review and Analysis of Vertical Breakwater Failures. MAST G6-S/Project 2, wave impact loading on vertical structures, Franzius-Institute, 1992.Google Scholar
  68. [68]
    J. PAMIN. Gradient-dependent plasticity in numerical simulation of localization phenomena. Delft University Press, 1994.Google Scholar
  69. [69]
    C. V. PAO. The existence and stability of the solutions of nonlinear operator differential equations. Arch. Rat. Mech. Analysis, 35: 16–29, 1969.MathSciNetMATHGoogle Scholar
  70. [70]
    C. V. PAO and W. G. VOGT. On the stability of nonlinear operator differential equations, and applications. Arch. Rat. Mech. Analysis, 35: 30–46, 1969.MathSciNetMATHGoogle Scholar
  71. [71]
    A. PAZY. Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, 1983.Google Scholar
  72. [72]
    P. PERZYNA. The constitutive equations for rate sensitive plastic materials. Quart. Appl. Math., 20: 321–332, 1963.MathSciNetMATHGoogle Scholar
  73. [73]
    P. PERZYNA. Fundamental problems in viscoplasticity. In C.-S. Yih, editor, Advances in Applied Mechanics, volume 9, pages 243–377. Academic Press, 1966.Google Scholar
  74. [74]
    P. PERZYNA. Thermodynamic theory of viscoplasticity. In Advances in Applied Mechanics, volume 11, pages 313–354. Academic Press, 1971.Google Scholar
  75. [75]
    P. PERZYNA. Constitutive equations of dynamic plasticity. In D. R. J. Owen, E. O nate, and E. Hinton, editors, Computational Plasticity, Fundamentals and Applications,pages 483–508, Swansea, 1992. Barcelona, April 6–10,Pineridge Press.Google Scholar
  76. [76]
    P. PERZYNA. Analysis of the fundamental equations describing thermoplastic flow process in solid body. Arch. Mech., 43: 287–296, 1993.Google Scholar
  77. [77]
    P. PERZYNA. Instability phenomena and adiabatic shear band localization in thermoplastic flow processes. Acta. Mech., 94: 1–31, 1994.MathSciNetGoogle Scholar
  78. [78]
    P. PERZYNA. Constitutive modelling of dissipative solids for localization and fracture (single cristals and polycristalline solids). Proceedinngs of Advanced School on Localization and fracture phenomena in inelastic solids, Udine (this volume), 1997.Google Scholar
  79. [79]
    S. PIETRUSZCZAK and Z. MRÔZ. Numerical analysis of elastic—plastic compression of pillars accounting for material hardening and softening. Int. J. Rock Mech. Min. Sci. Geomech., 17: 199–207, 1980.CrossRefGoogle Scholar
  80. [80]
    S. PIETRUSZCZAK and Z. MRüZ. Finite element analysis of deformation of strainüsoftening materials. Int. J. Num. Meth. in Eng.,17:327–334, 1981.Google Scholar
  81. [81]
    M. RENARDY and R.C. ROGERS. An Introduction to Partial Differential Equations. Springer-Verlag, New York, 1992.Google Scholar
  82. [82]
    J. R. RICE. The localization of plastic deformation. In W. T. Koiter, editor, Theoretical and Applied Mechanics, pages 207–220. North-Holland Publishing Company, 1976.Google Scholar
  83. [83]
    J. W. RUDNICKI and J. R. RICE. Conditions for the localization of deformations in pressure-sensitive dilatant meterials. J. Mech. Phys. of Solids, 23: 371–394, 1975.CrossRefGoogle Scholar
  84. [84]
    L. J. SLUYS. Wave propagation, localisation and dispersion in softening solids. Dissertation, Delft University of Technology, Department of Civil Engineering, 1992.Google Scholar
  85. [85]
    L. J. SLUYS. Different regularization methods for solution of the initial boundary value problems (geotechnical materials). Proceedinngs of Advanced School on Localization and fracture phenomena in inelastic solids, Udine (this volume), 1997.Google Scholar
  86. [86]
    L. J. SLUYS, J. BLOCK, and R. de BORST. Wave propagation and localization in viscoplastic media. In E. Hinton D. Owen, E. Orate, editor, III Int. Conf. on Comp. Plasticity, Fundamentals and Applications, COMPLAS III, Barcelona, Spain, April.1–9, 1992, pages 539–550, 1992.Google Scholar
  87. [87]
    P. STEINMANN and K. WILLAM. Localization within the framework of micropolar elasto-plasticity. In Advances in Continuum Mechanics. Springer, 1991.Google Scholar
  88. [88]
    H. P. STUWE. Experimental aspects of crystal plasticity. Proceedinngs of Advanced School on Localization and fracture phenomena in inelastic solids, Udine (this volume), 1997.Google Scholar
  89. [89]
    F. TAVENAS and S. LEROUEIL. The behaviour of embankments on clay foundations. Canadian Geotechnical Journal, 17: 236–260, 1980.CrossRefGoogle Scholar
  90. [90]
    F. TAVENAS, S. LEROUEIL, P. La ROCHELLE, and M. ROY. Creep behaviour of an undisturbed lightly overconsolidated clay. Canadian Geotechnical Juornal, 15: 402–423, 1978.CrossRefGoogle Scholar
  91. [91]
    A. TIKHONOV and V. AR.SENIN. Methods for the solution of incorrect problems (in Russian). Nauka, Moscow, 1979.Google Scholar
  92. [92]
    C. TRUESDELL and W. NOLL. The nonlinear field theories. In Handbuch der Physik, Band I11/3. Springer, Berlin, Heidelberg, New York, 1965.Google Scholar
  93. [93]
    Z. WASZCZYSZYN. Computational methods and plasticity. Report lr-583, TU Delft, 1989.Google Scholar
  94. [94]
    G. WEBER and L. ANAND. Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoplastic solids. Comp. Meth. in Appl. Mech. Eng.,79:173–202, 1990.Google Scholar
  95. [95]
    G. WEBER, A. M. LUSH, A. ZAVALANGOS, and L. ANAND. An objective time-integration procedure for isotropic rate-independent and rate-dependent elastic-plastic constitutive equations. Int.J. of Plasticity, 6: 701–744, 1990.CrossRefMATHGoogle Scholar
  96. [96]
    D. M. WOOD. Soil Behaviour and Critical State Soil Mechanics. Technical report, Cambridge University Press, 1990.MATHGoogle Scholar
  97. [97]
    N. ZABARAS and A. F. M. ARIF. A family of integration algorithms for constitutive equations in in finite deformation elasto-viscoplasticity. Int. J. Num. Meth. in Eng.,33:59–84, 1992.Google Scholar
  98. [98]
    H. M. ZBIB and J. S. JUBRAN. Dynamic sheat banding: A three—dimensional analysis. Int. Jour. of Plasticity, 8: 619–641, 1992.CrossRefGoogle Scholar
  99. [99]
    M. ŻYCZKOWSKI. Combined Loadings in the Theory of Plasticity. PWN-Polish Scientific Publishers, 1981.Google Scholar

Copyright information

© Springer-Verlag Wien 1998

Authors and Affiliations

  • T. Lodygowski
    • 1
  1. 1.Technical University of PoznanPoznanPoland

Personalised recommendations