Endochronic inelasticity and incremental plasticity

Zdenk P. Bazant*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

108 Scopus citations


Endochronic and non-associated plastic formulations are compared by introducing an "inelastic stiffness locus", defined as the locus of all strain increments in the strain space which give the same magnitude of inelastic strain increments. For classical plasticity the locus is a straight line, while for endochronk formulations it is a circle, sphere or a quadratic surface (ellipsoid). Similarly to vertex hardening models and the deformation theory of plasticity, endochronic theory gives inelastic strain for strain increments tangential to the current loading surface, while plasticity gives perfectly elastic response. However, in contrast to vertex hardening, the endochronic inelastic strain for tangential strain increments is normal to the loading surface. Consequently, endochronic theory is stiffer than vertex hardening for this loading direction and is less prone to indicate instability. However, it is softer than plasticity. Among all possible constitutive relations, plasticity (without yield vertex) is least prone to indicate material instability, and so it is the least safe model to assume if test data are inconclusive as far as the type of constitution law is concerned. Tangential linearization of the endochronic inelasticity is presented. The tensor of tangential moduli, with all of its components, depends continuously on the strain increment direction in the strain space. Endochronic analogs of the loading surface and of kinematic and isotropic hardening rules are indicated, and stress-induced anisotropy of the quadratic form defining intrinsic time increments is formulated. It is shown that for proportional loading an endochronic formulation can be readily converted to an equivalent plasticity formulation. The fracturing material theory in which the loading function depends on strain rather than stress is also analyzed and it is shown that its inelastic stiffness locus is similar as for plasticity. Implications for material instability, and especially for stability of the response to pulsating loads of small amplitude, are discussed. By contrast to plasticity, but similarly to viscoplasticity, the endochronic inelasticity violates Liapunov-type stability conditions, but it meets a proper continuity condition. Refinements to satisfy both are possible, but questionable if one deals with materials such as geological materials, which are unstable or exhibit strain softening. Introducing unloading and reloading criteria and a certain type of kinematic hardening, the endochronic formulation may be refined so as to model cyclic strain accumulation yet satisfy Drucker's postulate for the hysteresis loops.

Original languageEnglish (US)
Pages (from-to)691-714
Number of pages24
JournalInternational Journal of Solids and Structures
Issue number9
StatePublished - 1978

ASJC Scopus subject areas

  • Modeling and Simulation
  • General Materials Science
  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics


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