Interaction between heat dipole and circular interfacial crack

  • Wan-shen Xiao (肖万伸)Email author
  • Chao Xie (谢超)
  • You-wen Liu (刘又文)


The heat dipole consists of a heat source and a heat sink. The problem of an interfacial crack of a composite containing a circular inclusion under a heat dipole is investigated by using the analytical extension technique, the generalized Liouville theorem, and the Muskhelishvili boundary value theory. Temperature and stress fields are formulated. The effects of the temperature field and the inhomogeneity on the interfacial fracture are analyzed. As a numerical illustration, the thermal stress intensity factors of the interfacial crack are presented for various material combinations and different positions of the heat dipole. The characteristics of the interfacial crack depend on the elasticity, the thermal property of the composite, and the condition of the dipole.

Key words

thermoelasticity heat dipole interfacial crack circular inclusion inhomogeneity 



inclusion radius


interfacial crack arc


interface arc

S+, S

inclusion domain and matrix domain


material numbers for inclusion and matrix

μj, κj

elasticity constants of the jth material


polar radius of heat dipole center location


polar angle of heat dipole center location


half span length of heat dipole

z1, z2

coordinates of heat source and heat sink


angle between the span line and the x-axis


half of the central angle of the crack arc


temperature field function


real part of a complex variable or a complex function


imaginary unit

\( \overline {( \cdot )} \)

conjugate value of a complex parameter


the real part of its differential is the complex function for temperature


heat flux


total heat transfer rate


the coefficient of heat conductivity


normal stress


shear stress

ϕ(·), ψ(·)

complex potential functions for stress

Φ(·), Ψ(·)

differential function of complex potential for stress


heat expansion coefficient

u, υ


e, f, K, W, σ, S, J, N, O


Kj, Kjl

stress intensity factors of crack tip

Y (·)

Plemelj function


transform function

Chinese Library Classification

TB381 O343.7 

2000 Mathematics Subject Classification

74R10 74F05 


  1. [1]
    Mindlin, R. D. and Cheng, D. H. Thermoelastic stress in the semi-infinite solid. Journal of Applied Physics 21(9), 931–933 (1950)zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Zhu, Z. H. and Muguid, S. A. On the thermoelastic stresses of multiple interacting inhomogeneities. International Journal of Solids and Structures 37(16), 2313–2330 (2000)zbMATHCrossRefGoogle Scholar
  3. [3]
    Muskhelishvili, N. I. Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Leyden (1975)zbMATHGoogle Scholar
  4. [4]
    Chao, C. K. and Chang, R. C. Thermal interface crack problems in dissimilar anisotropic media. Journal of Applied Physics 72(7), 2598–2604 (1992)CrossRefGoogle Scholar
  5. [5]
    Qin, Q. H. Thermoelectroelastic solution for elliptic inclusions and application to crack-inclusion problems. Applied Mathematical Modelling 25(1), 1–23 (2000)zbMATHCrossRefGoogle Scholar
  6. [6]
    Xiao, W. S. and Wei, G. Interaction between screw dislocation and circular crack under uniform heat flux (in Chinese). Journal of Mechanical Strength 29(5), 779–783 (2007)Google Scholar
  7. [7]
    Pham, C. V., Hasebe, N., Wang, X. F., and Saito, T. Interaction between a cracked hole and a line crack under uniform heat flux. International Journal of Fracture 13(4), 367–384 (2005)Google Scholar
  8. [8]
    Hasebe, N., Wang, X. F., Saito, T., and Sheng, W. Interaction between a rigid inclusion and a line crack under uniform heat flux. International Journal of Solids and Structures 44(7–8), 2426–2441 (2007)zbMATHCrossRefGoogle Scholar
  9. [9]
    Chao, C. K. and Shen, M. H. On bonded circular inclusions in plane thermoelasticity. Journal of Applied Mechanics 64(4), 1000–1004 (1997)zbMATHCrossRefGoogle Scholar
  10. [10]
    Chao, C. K. and Tan, C. J. On the general solutions for annular problems with a point heat source. Journal of Applied Mechanics 67(3), 511–518 (2000)zbMATHGoogle Scholar
  11. [11]
    Rahman, M. The axisymmetric contact problem of thermoelasticity in the presence of an internal heat source. International Journal of Engineering Science 41(16), 1899–1911 (2003)CrossRefMathSciNetGoogle Scholar
  12. [12]
    Chao, C. K. and Chen, F. M. Thermal stresses in an isotropic trimaterial interacted with a pair of point heat source and heat sink. International Journal of Solids and Structures 41(22–23), 6233–6247 (2004)zbMATHCrossRefGoogle Scholar
  13. [13]
    Hasebe, N. and Wang, X. F. Complex variable method for thermal stress problem. Journal of Thermal Stresses 28(6–7), 595–648 (2005)CrossRefMathSciNetGoogle Scholar
  14. [14]
    Sih, G. C., Raris, P. C., and Erdogan, F. Crack-tip stress factors for plane extension and plane bending problem. Journal of Applied Mechanics 29(1), 306–312 (1962)Google Scholar

Copyright information

© Shanghai University and Springer Berlin Heidelberg 2009

Authors and Affiliations

  • Wan-shen Xiao (肖万伸)
    • 1
    Email author
  • Chao Xie (谢超)
    • 1
  • You-wen Liu (刘又文)
    • 1
  1. 1.College of Mechanics and AerospaceHunan UniversityChangshaP. R. China

Personalised recommendations