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Thermoelastic equilibrium of piecewise homogeneous solids with thin inclusions

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Abstract

An effective method of modeling the presence of thin inclusions of arbitrary physical nature in bodies is discussed. Using this method, the plane thermoelastic problem for two bounded dissimilar semi-planes with thin heat-active interface inclusions is reduced to two separate systems of singular integral equations. The concept of generalized stress-intensity factors is introduced and their dependence on the material characteristics and several methods of thermal loading are analyzed.

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Abbreviations

PCCDD:

The principle of conjugation of continua with different dimensions

JF:

Jump functions

CPMF:

Constituents of physical and mechanical fields

IC:

Interaction conditions

SIE:

System of integral equations

SSIE:

System of singular integral equations

HII:

Heat-insulated inclusion

DI:

Diathermic inclusion

GTFIF:

Gradients of the temperature-field intensity factors

GSIF:

Generalized stress-intensity factors

f r :

Jump functions

T :

Temperature

\({{T_{,n}\equiv \partial T}/{\partial n},}\) :

 

q x , q y , q n :

Heat fluxes

σ kj (ξ), u k (ξ):

Stresses, displacements

\({{L}'_p = [a_p^- ;\;a_p^+]}\) :

Line, modeling the presence of a thin inclusion

2h (x):

Inclusion width

δ :

Plate thickness

q B (x,y,z):

Specific density of heat sources

E :

Young’s modulus

ν :

Poisson’s ratio

α T :

Coefficient of thermal expansion of the material

λ k , λ B :

Heat-conduction coefficients

α yk :

Coefficients of heat emission or values reverse to coefficients of heat resistance

N x (w), N xy (w)U(w), V(w):

Stresses and displacements at the inclusion tips

Q x (w ):

Heat flux at the inclusion tips

M (w ):

Moments at the inclusion tips

\({\varepsilon _{\rm B}^w }\) :

Rigid turn of the inclusion

Φ T (z ):

Complex potential of the temperature field

\({\Phi _k \left( z \right),\;\Psi _k \left( z \right)}\) :

Complex Kolosov–Muskhelishvili potentials

\({k_1^\pm ,k_2^\pm }\) :

Gradients of the temperature-field intensity factors

\({K_{1,m}^\pm ,\;K_{2,m}^\pm \;}\) :

Generalized stress-intensity factors

\({\begin{array}{l}\left\langle \varphi \right\rangle_{h}=\varphi\left( x,-h \right)+\varphi(x,h),\\ \left[\varphi \right]_{h}=\varphi\left( x,-h \right)-\varphi(x,h)\end{array}}\) :

\({\begin{array}{l}\left[ \varphi \right]=\varphi _1^- \left( x \right)-\varphi _2^+\left( x \right),\\ \left\langle \varphi \right\rangle=\varphi _1^- \left( x \right)+\varphi _2^+\left( x \right)\end{array}}\)

superscripts “ + ”, “ − ”:

Denotes the boundary values of the functions on the upper and the lower inclusion borders with respect to width

a hat “^":

Marks the disturbed constituents of the fields

superscript “o”:

Marks the CPMF in the corresponding problem without any inclusion

subscript “B”:

Denotes the terms CPMF inside the inclusion

superscript “b”:

Marks the biharmonic part of CPMF in the case of the generalized plane temperature field.

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Authors and Affiliations

  1. Lviv University, Universytetska 1 Street, Lviv, 79000, Ukraine

    G. T. Sulim

  2. Bialystok Technical University, Wiejska 45A Street, Bialystok, 15351, Poland

    G. T. Sulim

  3. Ukrainian Academy of Printing, Pidgolosko 19 Street, Lviv, 79020, Ukraine

    J. Z. Piskozub

Authors
  1. G. T. Sulim
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  2. J. Z. Piskozub
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Correspondence to J. Z. Piskozub.

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Sulim, G.T., Piskozub, J.Z. Thermoelastic equilibrium of piecewise homogeneous solids with thin inclusions. J Eng Math 61, 315–337 (2008). https://doi.org/10.1007/s10665-008-9225-3

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