Continuum Mechanics and Thermodynamics

, Volume 29, Issue 1, pp 251–269 | Cite as

Constitutive equations of a tensorial model for strain-induced damage of metals based on three invariants

  • Nikolai D. Tutyshkin
  • Paul LofinkEmail author
  • Wolfgang H. Müller
  • Ralf Wille
  • Oliver Stahn
Original Article


On the basis of the physical concepts of void formation, nucleation, and growth, generalized constitutive equations are formulated for a tensorial model of plastic damage in metals based on three invariants. The multiplicative decomposition of the metric transformation tensor and a thermodynamically consistent formulation of constitutive relations leads to a symmetric second-order damage tensor with a clear physical meaning. Its first invariant determines the damage related to plastic dilatation of the material due to growth of the voids. The second invariant of the deviatoric damage tensor is related to the change in void shape. The third invariant of the deviatoric tensor describes the impact of the stress state on damage (Lode angle), including the effect of rotating the principal axes of the stress tensor (Lode angle change). The introduction of three measures with related physical meaning allows for the description of kinetic processes of strain-induced damage with an equivalent parameter in a three-dimensional vector space, including the critical condition of ductile failure. Calculations were performed by using experimentally determined material functions for plastic dilatation and deviatoric strain at the mesoscale, as well as three-dimensional graphs for plastic damage of steel DC01. The constitutive parameter was determined from tests in tension, compression, and shear by using scanning electron microscopy, which allowed to vary the Lode angle over the full range of its values Open image in new window . In order to construct the three-dimensional plastic damage curve for a range of triaxiality parameters \(-1 \le ST \le 1\) and of Lode angles Open image in new window , we used our own, as well as systematized published experimental data. A comparison of calculations shows a significant effect of the third invariant (Lode angle) on equivalent damage. The measure of plastic damage, based on three invariants, can be useful for assessing the quality of metal mesostructure produced during metal forming processes. In many processes of metal sheet forming the material experiences, a non-proportional loading accompanied by rotating the principal axes of the stress tensor and a corresponding change of Lode angle.


Void Plasticity Damage Ductile failure Constitutive equations Invariants 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Construction, Constructional Materials and StructuresTula State UniversityTulaRussia
  2. 2.Chair of Continuum Mechanics and Materials Theory, Institute of MechanicsTechnische Universität BerlinBerlinGermany

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