Volume 2, Issue 1, No.2 PDF DOWNLOAD
  • Title:
  • Numerical solution of dissimilar inclusion problem for square notch with small round corners in plane elasticity
  • Author:

    Yizhou Chen

  • Author Affiliation:

    Division of Engineering Mechanics, Jiangsu University, Zhenjiang, P. R. China

  • Received:Oct.24, 2022
  • Accepted:Nov.22, 2022
  • Published:Dec.26, 2022

In this paper, the stress solution for a square notch with small round corners is studied. The square notch is composed of two portions, or the matrix and the inclusion. The elastic property of the inclusion portion is different to that of the matrix. Remote loading is applied. When the dissimilar inclusion is embedded in the matrix, the stress state in the inclusion will be changed significantly. The interface conditions along interface are formulated exactly.  Complex variable function method is used to solve the problem. The complex potentials in the matrix and the inclusion are assumed in different structure.  In the formulation, the number of the equations is larger than that of the unknowns. Therefore, the weight residue technique is used to solve the algebraic equation. The tangential, normal and shear stress components along the interface from the matrix and inclusion side are evaluated from the solution of the algebraic equation. In the computation, significant stress concentration has been found for the tangential stress component.


Dissimiler inclusion, inhomogeneity, complex variable method, weight residue technique, stress concentration factor


[1] N.I. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity, Noordhoof, Groningen, 1963.

[2] S.N. Savin, Stress Distribution around Notches (English translation edition), Pergamon Press, Oxford, 1961.

[3] T. Mura, Micromechanics of Defects in Solids, Second ed. Martinus Nijhoff, Dordrecht, Netherlands, 1987.

[4] K. Zhou, H.J. Hoh, X. Wang, L.M. Keer, J.H.L. Pang, B. Song, Q.J. Wang, A review of recent works on inclusions, Mech. Mater. 60 (2013) 144-158.

[5] X.Q. Jin, Z.J. Wang, Q.H. Zhou, L.M. Keer, Q. Wang, On the solution of an elliptical inhomogeneity in plane elasticity by the equivalent inclusion method, J. Elast. 114 (2014) 1–18.

[6] C.S. Chang, H.D. Conway, Stress analysis of an infinite plate containing an elastic rectangular inclusion, Acta Mech. 8 (1969)160-173.

[7] C.Y. Dong, S.H. Lo, Y.K. Cheung, Stress analysis of inclusion problems of various shapes in an infinite anisotropic elastic medium, Comput. Methods Appl. Mech. Engrg. 192 (2003) 683–696.

[8] M.H. Shen,· S.N. Chen,· F.M. Chen, Antiplane study on confocally elliptical inhomogeneity problem using an alternating technique, Arch. Appl. Mech. 75 (2016) 302–314.

[9] J. T. Chen, A. C. Wu, Null-field approach for the multi-inclusion problem under antiplane shears, ASME J. Appl. 74 (2007) 469- 487. 8

[10] J. C. Luo, C. F. Gao, Faber series method for plane problems of an arbitrarily shaped inclusion, Acta Mech. 208 (2009) 133– 145.

[11] X. Wang, Three-phase elliptical inclusions with internal uniform hydrostatic stresses in finite plane elastostatics, Acta Mech. 219 (2011) 77-90.

[12] X. Wang, X.L. Gao, On the uniform stress state inside an inclusion of arbitrary shape in a three-phase composite, Z. Angew. Math. Phys. 62 (2011) 1101–1116.

[13] Y.Z. Chen, Transfer matrix method for the solution of multiple elliptic layers with different elastic properties, Part. 1. Acta Mech. 226 (2015) 191-229.

[14] Y.Z. Chen, Solution for square notch problem with small round corners in plane elasticity under remote loading, Arch. Appl. Mech. 2021; 91: 2943-2947

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