Fractional calculus derivation of a rate-dependent PPR-based cohesive fracture model: theory, implementation, and numerical results

Oliver Giraldo-Londoño, Glaucio H. Paulino, William G. Buttlar

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


Rate-dependent fracture has been extensively studied using cohesive zone models (CZMs). Some of them use classical viscoelastic material models based on springs and dashpots. However, such viscoelastic models, characterized by relaxation functions with exponential decay, are inadequate to simulate fracture for a wide range of loading rates. To improve the accuracy of existing models, this work presents a mixed-mode rate-dependent CZM that combines the features of the Park-Paulino-Roesler (PPR) cohesive model and a fractional viscoelastic model. This type of viscoelastic model uses differential operators of non-integer order, leading to power-law-type relaxation functions with algebraic decay. We derive the model in the context of damage mechanics, such that undamaged viscoelastic tractions obtained from a fractional viscoelastic model are scaled using two damage parameters. We obtain these parameters from the PPR cohesive model and enforce them to increase monotonically during the entire loading history, which avoids artificial self-healing. We present three examples, two used for validation purposes and one to elucidate the physical meaning of the fractional differential operators. We show that the model is able to predict ratedependent fracture process of rubber-like materials for awide range of loading rates and that it can capture ratedependent mixed-mode fracture processes accurately. Results from the last example indicate that the order of the fractional differential operators acts as a memorylike parameter that allows for the fracture modeling of long-and short-term memory processes. The ability of fractional viscoelastic models tomodel this type of process suggests that relaxation functions with algebraic decay lead to accurate fracture modeling of materials for a wide range of loading rates.

Original languageEnglish (US)
Pages (from-to)1-29
Number of pages29
JournalInternational Journal of Fracture
Issue number1
StatePublished - Mar 2019
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mechanics of Materials
  • Computational Mechanics
  • Modeling and Simulation


  • Cohesive zone model
  • Damage mechanics
  • Fractional calculus
  • Fractional differential equations
  • Fractional viscoelasticity
  • Fracture mechanics
  • Rate-dependent fracture


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