Summary
Calculations of stresses in an infinite plane, containing two circular holes, can be performed by means of bipolar coordinates. In the case considered, the stresses are caused by the difference in temperature of the boundaries of the holes. A method is given to calculate the stresses and the strains. The calculation of the stresses is performed for a particular case.
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Abbreviations
- T :
-
temperature,
- ±T 0 :
-
temperature at the boundaries
- x,y:
-
rectangular coordinates
- \(\begin{array}{l} \Delta _0 = \frac{{\partial ^2 }}{{\partial x^2 }} + \frac{{\partial ^2 }}{{\partial y^2 }}, \\ \sigma _{xx} , \sigma _{xy} , \sigma _{yy} \\ \end{array}\) :
-
stresses in rectangular coordinates
- ε xx ,ε xy ,ε yy :
-
strains in rectangular coordinates
- ξ, η:
-
bipolar coordinates
- \(\begin{array}{l} \Delta = \frac{{\partial ^2 }}{{\partial \xi ^2 }} + \frac{{\partial ^2 }}{{\partial \eta ^2 }}, \\ \sigma _{\xi \xi } , \sigma _{\xi \eta } , \sigma _{\eta \eta } \\ \end{array}\) :
-
stresses in bipolar coordinates
- ε ξξ ,ε ξη ,ε ηη :
-
strains in bipolar coordinates
- G :
-
shear modulus
- μ:
-
Poisson’s ratio
- α:
-
linear coefficient of expansion
- p,q:
-
displacements in rectangular coordinates
- u,v:
-
displacements in bipolar coordinates
- ψ:
-
potential of displacement
- χ:
-
Airy stress function
- ξ 0 :
-
ξ boundary
- f :
-
−2Gψ + χ
- z :
-
ch g − cos η
- g :
-
zf
- a :
-
parameter used for the transformation of coordinates
- k :
-
integer
- A k ,B k :
-
constants
- C :
-
a2 (1 + μ)αT0G/ξ0
- D,F,H,L:
-
constants
- P,Q:
-
auxiliary functions for the calculation of displacements
- r :
-
radius
- y m :
-
ordinate of centre
- α k ,β k :
-
vide (32)
References
Jeffery, G. B., Phil. Trans. Roy. Soc. U221 (1920) 265.
Me Ian, E. und H. Parkus, Wärmespannungen, Springer, Wien, 1953; Abschnitt V.
Biezeno, C. B. und R. Grammel. Technische Dynamik, Spinger, Berlin, 1939; T1 I, 18.
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Van Der Linden, C.A.M. Thermal stresses in a plate containing two circular holes of eoual radius, the boundaries of which are kept at different tempatures. Appl. sci. Res. 6, 117–128 (1956). https://doi.org/10.1007/BF03185031
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DOI: https://doi.org/10.1007/BF03185031