Elasticity_ Barber

358 22 Thermoelastic displacement potentials
by superposing solutions A,B,T and taking
φ = 2(1 − ν)ψ; ω = −χ = ∂ψ
∂z . (22.57)
The resulting expressions are given in Table 22.1 as Solution P. A striking
result is that the three stress components σ zx , σ zy , σ zz are zero, not merely on
the plane z =0, where we have forced them to zero by the relations (22.57), but
throughout the half-space. For this reason, the solution is also appropriate to
the problem of a thick plate a<z <b with traction free faces. Because of this
feature, the solution is henceforth referred to as thermoelastic plane stress.
Notice also the similarity of the expressions for the non-zero stresses
σ xx , σ xy , σ yy to the corresponding expressions obtained from the Airy stress
function. However, the present solution is not two-dimensional — i.e. the
temperature and hence all the non-zero stress and displacement components
can be functions of all three coördinates — and it is exact, whereas the twodimensional
plane stress solution is generally approximate.
Another result of interest is
∂ 2 u z
∂x 2
+ ∂2 u z − ν)
= −(1
∂y2 µ
( ∂ 3 ψ
∂x 2 ∂z +
(because ψ and hence ∂ψ/∂z is harmonic)
)
∂3 ψ
∂y 2 =
∂z
(1 − ν) ∂ 3 ψ
µ ∂z 3
= α(1 + ν)q z
K
= δq z , (22.58)
where δ is the thermal distortivity (see §14.3.1).
In other words, the sum of the principal curvatures of any distorted z-plane
is proportional to the local heat flux across that plane 4 . The corresponding
two-dimensional result was proved in §14.3.1.
PROBLEMS
1. The temperature T (R) in a hollow sphere b < R < a depends only on
the radius R and the surfaces of the sphere are traction-free. Use a method
similar to that in §22.1.1 to obtain expressions for the stress and displacement
components for any function T (R).
Note: The most general spherically symmetrical isothermal state of stress
(i.e. the general homogeneous solution) can be written by superposing (i) the
function A/R in solution A of Table 21.2 and (ii) a uniform state of hydrostatic
stress. The identity
4 Further consequences of this result are discussed in J.R.Barber, Some implications
of Dundurs’ theorem for thermoelastic contact and crack problems, Journal of
Strain Analysis, Vol. 22 (1980), pp.229–232.