Journal of Elasticity

, Volume 134, Issue 2, pp 235–241 | Cite as

Integral-Type Stress Boundary Condition in the Complete Gurtin-Murdoch Surface Model with Accompanying Complex Variable Representation

  • Ming Dai
  • Yong-Jian Wang
  • Peter SchiavoneEmail author


In the large majority of papers utilizing the Gurtin-Murdoch (G-M) model of a material surface, the complete model is avoided in favor of various modified versions often because they lead to simpler representations of the corresponding stress boundary condition. We propose in this paper an integral-type stress boundary condition for the plane deformations of a bulk-interface composite system which allows for an equally simple implementation of the complete G-M model. Since the mechanical behavior of such composite systems is often analyzed using complex variable methods, we formulate our ideas accordingly, in this context. Remarkably, in contrast to what is often believed to be the case, we find that boundary value problems based on our formulation of the stress boundary condition offer no added difficulty when utilizing the complete G-M model versus its various simplified counterparts. This new representation of the stress boundary condition is concise in form and will prove to be extremely useful in, for example, the calculation of the elastic field in the vicinity of nano-inhomogeneities of irregular shape.


Gurtin-Murdoch model Complex variable methods Surface/interface stress Nano-inhomogeneity 

Mathematics Subject Classification

34B05 74B05 74G05 



Wang thanks the National Natural Science Foundation of China (Grant No.: 11702147). Schiavone thanks the Natural Sciences and Engineering Research Council of Canada for their support through a Discovery Grant (Grant No.: RGPIN-2017-03716115112).


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.State Key Laboratory of Mechanics and Control of Mechanical StructuresNanjing University of Aeronautics and AstronauticsNanjingChina
  2. 2.School of Mechanical EngineeringChangzhou UniversityChangzhouChina
  3. 3.College of EngineeringNanjing Agricultural UniversityNanjingChina
  4. 4.Department of Mechanical EngineeringUniversity of AlbertaEdmontonCanada

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