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A Stress-Based Variational Model for Ductile Porous Materials and Its Extension Accounting for Lode Angle Effects

  • Long ChengEmail author
  • Vincent Monchiet
  • Géry de Saxcé
  • Djimédo Kondo
Chapter

Abstract

The aim of this work is to derive by homogenization techniques a macroscopic plastic model for porous materials with von Mises matrix. In contrast to the Gurson’s well known kinematical approach [19] applied to a hollow sphere, the proposed study proceeds by means of a statical limit analysis procedure, for which a suitable trial stress field is proposed. In the first part, the formulation of the stress variational model is developed, by considering the Hill’s variational principle, and introducing a Lagrange’s multiplier to solve the resulting saddle-point minimization problem. This methodology being opposite to the Gurson’s kinematical approach, complements the limit analysis methods for porous materials. The second part is devoted to an application of the proposed approach to the porous materials with von Mises matrix. To this end, an axisymmetric model is first studied by adopting a suitable trial stress field, which is composed by a heterogeneous part corresponding to the exact solution of hydrostatic loading and a homogeneous part for capturing the shear effects. We derive closed form formula which depends not only on the first and second invariant of the macroscopic stress tensor but also on the sign of the third invariant of the stress deviator. Moreover, an extension of the above axisymmetric model to the general case of non-axisymmetric loadings by introducing a more general trial stress field is studied. The established new yield locus explicitly depends on the effect of the third invariant (equivalently the Lode angle). The obtained results are fully discussed and compared to existing models, available numerical data and to Finite Elements results obtained from cell calculation carried out during the present study.

Keywords

Yield Surface Stress Deviator Hollow Sphere Stress Triaxiality Lode Angle 
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.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Long Cheng
    • 1
    Email author
  • Vincent Monchiet
    • 2
  • Géry de Saxcé
    • 1
  • Djimédo Kondo
    • 3
  1. 1.Laboratoire de Mécanique de Lille (UMR CNRS 8107)Université de Lille 1, Cité ScientifiqueVilleneuve d’AscqFrance
  2. 2.Laboratoire Modlisation et Simulation Multi Echelles (UMR CNRS 8208)Université Paris-EstMarne la ValléeFrance
  3. 3.Institut Jean Le Rond d’Alembert (UMR CNRS 7190)Université Pierre et Marie CurieParisFrance

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