International Journal of Fracture

, Volume 216, Issue 1, pp 1–29 | Cite as

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

  • Oliver Giraldo-Londoño
  • Glaucio H. PaulinoEmail author
  • William G. Buttlar
Original Paper


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 rate-dependent fracture process of rubber-like materials for a wide range of loading rates and that it can capture rate-dependent mixed-mode fracture processes accurately. Results from the last example indicate that the order of the fractional differential operators acts as a memory-like parameter that allows for the fracture modeling of long- and short-term memory processes. The ability of fractional viscoelastic models to model 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.


PPR cohesive zone model Fractional viscoelasticity Fractional calculus Fractional differential equations Rate-dependent fracture Damage mechanics Fracture mechanics 



The authors gratefully acknowledge support by the Raymond Allen Jones Chair at the Georgia Institute of Technology. The interpretation of the results of this work is solely that by the authors, and it does not necessarily reflect the views of the sponsors or sponsoring agencies.


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Copyright information

© Springer Media B.V., onderdeel van Springer Nature 2019

Authors and Affiliations

  • Oliver Giraldo-Londoño
    • 1
  • Glaucio H. Paulino
    • 1
    Email author
  • William G. Buttlar
    • 2
  1. 1.School of Civil and Environmental EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.School of Civil and Environmental EngineeringUniversity of MissouriColumbiaUSA

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