pp 1–18 | Cite as

‘Explicit’ and ‘implicit’ non-local continuous descriptions for a plate with circular inclusion in tension

  • Meral TunaEmail author
  • Lorenzo Leonetti
  • Patrizia Trovalusci
  • Mesut Kirca
Computational Models for 'Complex' Materials and Structures, beyond the Finite Elements


Increasing application of composite structures in engineering field inherently speed up the studies focusing on the investigation of non-homogeneous bodies. Due to their capability on capturing the size effects, and offering solutions independent of spatial discretization, enriched non-classical continuum theories are often more preferable with respect to the classical ones. In the present study, the sample problem of a plate with a circular inclusion subjected to a uniform tensile stress is investigated in terms of both ‘implicit’/‘weak’ and ‘explicit’/‘strong’ non-local descriptions: Cosserat (micropolar) and Eringen theories, by employing the finite element method. The material parameters of ‘implicit’ model is assumed to be known, while the nonlocality of ‘explicit’ model is optimized according to stress concentration factors reported for infinite Cosserat plates. The advantages/disadvantages, and correspondence/non-correspondence between both non-local models are highlighted and discussed apparently for the first time, by comparing the stress field provided for reference benchmark problem under various scale ratios, and material parameter combinations for matrix-inclusion pair. The results reveal the analogous character of both non-local models in case of geometric singularities, which may pave the way for further studies considering problems with noticeable scale effects and load singularities.


Non-local models Cosserat Eringen Nonhomogeneous solids Finite elements Scale effects 



This work was done when Meral Tuna was a Visiting Researcher at DISG, Sapienza University of Rome, with financial support of Italian Ministry of University and Research PRIN 2015, Project 2015JW9NJT (Grant No. B86J16002300001, ID: and PRIN 2017 (Grant No. B86J16002300001) “Materials with microstructure: multiscale models for the derivation non-local continua of ‘implicit/weak’ and explicit/strong’” which is gratefully acknowledged.

Compliance with ethical standards

Conflict of interest

The authors have no conflict of interest to report.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Meral Tuna
    • 1
    Email author
  • Lorenzo Leonetti
    • 2
  • Patrizia Trovalusci
    • 3
  • Mesut Kirca
    • 1
  1. 1.Department of Mechanical EngineeringIstanbul Technical UniversityIstanbulTurkey
  2. 2.Department of Structural EngineeringUniversity of CalabriaRendeItaly
  3. 3.Department of Structural and Geotechnical EngineeringSapienza - University of RomeRomeItaly

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