# Interaction between heat dipole and circular interfacial crack

## Abstract

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## Nomenclature

*R*inclusion radius

*L′*interfacial crack arc

*L*interface arc

*S*^{+},*S*^{−}inclusion domain and matrix domain

- I, II
material numbers for inclusion and matrix

*μ*_{j},*κ*_{j}elasticity constants of the

*j*th material*ρ*polar radius of heat dipole center location

*θ*polar angle of heat dipole center location

*r*half span length of heat dipole

*z*_{1},*z*_{2}coordinates of heat source and heat sink

*ϕ*angle between the span line and the

*x*-axis*α*half of the central angle of the crack arc

*T*(·)temperature field function

- Re(·)
real part of a complex variable or a complex function

- i
imaginary unit

- \( \overline {( \cdot )} \)
conjugate value of a complex parameter

*g*(·)the real part of its differential is the complex function for temperature

*q*heat flux

*Q*total heat transfer rate

*κ*_{t}the coefficient of heat conductivity

*σ*normal stress

*τ*shear stress

*ϕ*(·),*ψ*(·)complex potential functions for stress

- Φ(·), Ψ(·)
differential function of complex potential for stress

*β*heat expansion coefficient

*u, υ*displacements

*e, f, K, W, σ, S, J, N, O*constants

*K*_{j},*K*_{jl}stress intensity factors of crack tip

*Y*(·)Plemelj function

*ω*(·)transform function

## Chinese Library Classification

TB381 O343.7## 2000 Mathematics Subject Classification

74R10 74F05## References

- [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]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]Muskhelishvili, N. I.
*Some Basic Problems of the Mathematical Theory of Elasticity*, Noordhoff, Leyden (1975)zbMATHGoogle Scholar - [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]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]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]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]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]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]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]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]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]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]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