Journal of Failure Analysis and Prevention

, Volume 18, Issue 3, pp 607–618 | Cite as

Improvement of an Exponential Cohesive Zone Model for Fatigue Analysis

  • Wenlong Zhang
  • Ala Tabiei
Technical Article---Peer-Reviewed


Cohesive zone model is an important tool for fatigue analysis, especially for fatigue crack growth along with an interface. A pioneering model is the one of Roe and Siegmund (Eng Fract Mech 70:209–232, 2003), in which the damage accumulation is calculated using an irreversible exponential cohesive law. However, it is found in our recent research that the constant unloading and reloading slope in Roe’s damage evolution law could cause a discontinuity in the traction–separation curve when the mixed mode ratio changes. This limits its application to single mode cyclic loading or scenarios where the mixed mode ratio is constant. In this paper, the cause of such discontinuity is analyzed, and a robust cyclic loading formulation is proposed, which will help the exponential cohesive law remain continuous under arbitrary mixed mode cyclic loading. Moreover, it is found in this paper that by adding a scale factor to the Roe’s damage law, the fatigue failure time can be approximated with less computational cost. The relationship between fatigue failure time and the scale factor is shown to be inversely proportional.


Cohesive zone model Discontinuity Roe’s damage evolution law Fatigue Exponential cohesive law Cyclic loading 

List of symbols


Mixed mode ratio


The scale factor in proposed modified damage law


Endurance limit parameter in Roe’s damage law

dn, dt

Damage factor in the normal and tangential directions


Fatigue damage


Fatigue damage accumulation rate


Mixed mode separation threshold


The normal and tangential separations that correspond to the maximum traction


Normalization parameter for separation deformation rate


Mixed mode separation

\({\dot{\Delta }}\)

Separation deformation rate


Maximum mixed mode separation in a loading cycle

Δn, Δt

Normal and tangential separations in exponential cohesive law

Δn0, Δt0

Normal and tangential separations on the maximum separation envelope

Δn,max, Δt,max

Maximum separation in each loading cycle in the normal and tangential directions


The initial slope of exponential cohesive law in the normal and tangential directions


Number of loading cycles to fatigue failure with a scale factor C


Number of loading cycles to fatigue failure without a scale factor


Energy release rate


Mode I and mode II critical energy release rate


Load ratio


Normalization parameter for resultant traction


Minimum and maximum stress in cyclic loading


Mean stress in cyclic loading


Normal and tangential cohesive strength


Normal and tangential traction in cohesive law


Normal and tangential traction on the maximum separation envelope


Resultant traction


Fatigue failure time with a scale factor C


Fatigue failure time without a scale factor


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Copyright information

© ASM International 2018

Authors and Affiliations

  1. 1.Department of Civil EngineeringUniversity of CincinnatiCincinnatiUSA
  2. 2.Department of Mechanical EngineeringUniversity of CincinnatiCincinnatiUSA

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