Journal of Elasticity

, Volume 134, Issue 2, pp 175–191 | Cite as

Crack Growth Control Based on the Topological Derivative of the Rice’s Integral

  • M. Xavier
  • A. A. NovotnyEmail author
  • J. Sokołowski


In fracture mechanics, an important question concerns the useful life of mechanical components. Such components are, usually, submitted to the actions of external forces and/or degrading agents which can trigger the crack nucleation and propagation process. In particular, when a mechanical component is already partially cracked, the question is how to extend its remaining useful life. In this work, a simple and efficient methodology aiming to extend the remaining useful life of cracked elastic bodies is proposed. More precisely, we want to find a way to retard or even avoid the triggering of the crack propagation process by nucleating hard and/or soft inclusions far from the crack tip. The main idea consists in minimize a shape functional based on the Rice’s integral with respect to the nucleation of inclusions by using the concept of topological derivative. The obtained sensitivity, which corroborates with the famous Eshelby theorem, is used to indicate the regions where the controls have to be inserted. According to the Griffith’s energy criterion, this simple procedure allows for increasing the remaining useful life of the cracked body. Finally, some numerical experiments are presented showing the applicability of the proposed methodology.


Rice’s integral Griffith’s criterion Eshelby’s tensor Topological derivative Topology optimization 

Mathematics Subject Classification

74B05 74R10 49Q12 49J20 35C20 35A15 



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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Laboratório Nacional de Computação Científica LNCC/MCTCoordenação de Matemática Aplicada e ComputacionalPetrópolisBrazil
  2. 2.Institut Élie Cartan, UMR 7502 Laboratoire de MathématiquesUniversité de LorraineVandoeuvre Lès Nancy CedexFrance
  3. 3.Systems Research Institute of the Polish Academy of SciencesWarszawaPoland

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