Abstract
The thermoelastic effect has been used to study stress distributions in a number of in-plane loading problems. Analysis of the temperature distribution has been largely limited to isotropic one-dimensional approximations with heat transfer through the thickness of the specimen. In sonic fatigue, specimens undergo fully reversed bending with a stress gradient along the length of the specimen as well as through the thickness. This has also been modeled as a one-dimensional heat transfer problem with negligible heat transfer along the specimen length. The authors solve this as a two-dimensional problem for an isotropic material to determine the effect of heat transfer.
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Abbreviations
- A :
-
beam cross-sectional area (m2)
- C :
-
specific heat (J/kgK)
- D :
-
beam height (m)
- E :
-
Young's modulus (Pa)
- g(x, y, t), g*(x*, y*, t*):
-
volumetric and dimensionless volumetric heat generation (W/m3)
- g c :
-
dimensional constant from Newton's lawF=m a/g c (g c =1 kgm/Ns2)
- h :
-
convective heat transfer coefficient (W/m2K)
- H :
-
hD/k
- I :
-
second moment of area with respect to a centroidal axis (m4)
- k :
-
thermal conductivity (W/mK)
- L :
-
beam length (m)
- t, t* :
-
time (s) and dimensionless time
- t *max :
-
t * when the periodic surface temperatures is largest
- T :
-
absolute temperature (K)
- ΔT :
-
change in temperature due to thermoelastic effect
- ΔT exp :
-
experimentally measured surface ΔT
- ΔT max :
-
maximum surface ΔT (K)
- To :
-
initial temperature (K)
- T ∞ :
-
ambient temperature (K)
- T *,Tp* :
-
(T-T o)/ΔT max dimensionless temperature difference, period value ofT *
- T *max(x *):
-
maximum value ofT *(x*)=.5g 1*(x *)
- ΔU :
-
change in internal energy (J)
- V :
-
volume (m3)
- x, x* :
-
axial coordinate (m) and dimensionless axial coordinate
- y, y* :
-
vertical coordinate (m) and dimensionless vertical coordinate
- y L :
-
deflection at the tip of the beam,x=L (m)
- α:
-
k/ρ Cthermal diffusivity (m2/s)
- α i , α x , α y :
-
coefficient of thermal expansion in theith,x-andy-directions (K−1)
- β i :
-
ith eigenvalue for X(β i , x*
- Δσ i :
-
change in theith principal stress (Pa)
- ɛ xx :
-
normal strain in thex-(axial) direction
- γ j :
-
jth eigenvalue for Y(γ j , y*)
- η,η* :
-
constant (m−1) and dimensionless constant
- ρ:
-
density (kg/m3)
- ω,ω* :
-
fundamental frequency of beam (rad/s) and dimensionless frequency
References
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Wong, A.K., “A Non-adiabatic Thermoelastic Theory for Composite Laminates,”J. Phys. Chem. Solids,52,483–491 (1991).
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Byrd, L.W., Haney, M.A. Thermoelastic stress analysis applied to fully reversed bending fatigue. Experimental Mechanics 40, 10–14 (2000). https://doi.org/10.1007/BF02327542
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DOI: https://doi.org/10.1007/BF02327542