Archive of Applied Mechanics

, Volume 89, Issue 4, pp 737–754 | Cite as

Fatigue crack growth simulation in particulate-reinforced composites by the equivalent inclusion method and distributed dislocation method

  • Jiong Zhang
  • Zhan Qu
  • Weidong Liu
  • Li Zhou
  • Liankun WangEmail author


This paper presents an automatic fatigue crack propagation solution for particulate-reinforced composites. In this solution, the Eshelby’s equivalent inclusion method is used to solve the elastic fields of a two-dimensional plane containing multiple elliptical inclusions. Then the continuous distributed dislocations in an infinite plane are adopted to model the multiple cracks in the plane containing multiple elliptical inclusions. By combining the two methods, a system of singular integral equations can be formulated based on the stress condition of the cracks. The stress intensity factor of each crack can be obtained by solving the singular integral equations. The propagation of the cracks is studied based on the maximum circumferential stress criterion. Numerical examples are presented to show the complicated changes in the stress intensity factors and crack growth behaviors of the cracks affected by the inclusions.


Particulate-reinforced composites Distributed dislocations Inclusions Crack propagation 

List of symbols


Half length of the crack


Half length of the Ith crack

\(\sigma _{{{\textit{ij}}}}^0\)

Remote applied stress

\(K_I, K_{{\textit{II}}}\)

Mode I and mode II stress intensity factor

\(F_I, F_{{\textit{II}}}\)

Mode I and mode II non-dimensional stress intensity factor

\(N_{{\text {inc}}}\)

Total number of the elliptical inclusions in the plane


Elastic moduli of the Ith inclusion


Stress influence function in the local coordinates

\(\theta _I\)

Crack incline angle for the Ith crack in the global coordinates

\(\mu \)

Shear modulus of the elastic plane

\(\mu _i\)

Shear modulus of the ith inclusion

\(\nu \)

Poisson’s ratio

\(\kappa \)

Kolosov’s constant


Interior Eshelby’s tensor and exterior Eshelby’s tensor


Axes of the global coordinate system

\(\Delta a_i\)

Crack growth increments

\(\Delta a_{\max }\)

Maximum value of the crack propagation length


Fatigue propagation constant for the material

\(K_{{\text {Ieq}}}\)

Effective stress intensity factor

\({\mathop {\theta }\limits ^{\wedge }}_c\)

Crack turning angle

\(x_{{\text {tip}}}, y_{{\text {tip}}}\)

Location of the crack tip in the local coordinate system



This work was financially supported by National Natural Science Foundation of China (Nos. 51674200, 20873999) and Natural Science Foundation of Guangdong Province (No. 2018A030313430) and innovation and strong school engineering Foundation of Guangdong Province (Nos. 2017KQNCX201, 2016KQNCX169).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

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

Authors and Affiliations

  • Jiong Zhang
    • 1
  • Zhan Qu
    • 2
  • Weidong Liu
    • 3
  • Li Zhou
    • 1
  • Liankun Wang
    • 1
    Email author
  1. 1.School of Civil Engineering and ArchitectureWuyi UniversityJiangmenPeople’s Republic of China
  2. 2.School of Petroleum EngineeringXi’an Shiyou UniversityXi’anPeople’s Republic of China
  3. 3.College of Energy and Electrical EngineeringHohai UniversityNanjingPeople’s Republic of China

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