Continuum Mechanics and Thermodynamics

, Volume 30, Issue 5, pp 1125–1144 | Cite as

Quasi-brittle damage modeling based on incremental energy relaxation combined with a viscous-type regularization

  • K. Langenfeld
  • P. Junker
  • J. Mosler
Original Article


This paper deals with a constitutive model suitable for the analysis of quasi-brittle damage in structures. The model is based on incremental energy relaxation combined with a viscous-type regularization. A similar approach—which also represents the inspiration for the improved model presented in this paper—was recently proposed in Junker et al. (Contin Mech Thermodyn 29(1):291–310, 2017). Within this work, the model introduced in Junker et al. (2017) is critically analyzed first. This analysis leads to an improved model which shows the same features as that in Junker et al. (2017), but which (i) eliminates unnecessary model parameters, (ii) can be better interpreted from a physics point of view, (iii) can capture a fully softened state (zero stresses), and (iv) is characterized by a very simple evolution equation. In contrast to the cited work, this evolution equation is (v) integrated fully implicitly and (vi) the resulting time-discrete evolution equation can be solved analytically providing a numerically efficient closed-form solution. It is shown that the final model is indeed well-posed (i.e., its tangent is positive definite). Explicit conditions guaranteeing this well-posedness are derived. Furthermore, by additively decomposing the stress rate into deformation- and purely time-dependent terms, the functionality of the model is explained. Illustrative numerical examples confirm the theoretical findings.


Convexity Damage Rate-dependency Regularization Relaxation-based 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MechanicsTU DortmundDortmundGermany
  2. 2.Lehrstuhl für Mechanik - MaterialtheorieRuhr-Universität BochumBochumGermany

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