ABSTRACT
Displacement-controlled stresses can be calculated only in a limited number of cases. Figure 4.1 shows a number of examples illustrating the former type of stresses. Let us assume that the elastic plate has originally uniform temperature Tq quenched from one face to the Tf temperature. The relevant temperature distri bution and internal strain a,- are shown in Fig. 4.1. When one direction is restrained against expansion, a om compressive membrane stress must be applied:
°m = ~ l J E a (T - T0) dx' = Ea@m(T0 - Tf) -L/2
where
L = plate thickness E = Young’s modulus a = coefficient of linear expansion
0 m = nondimensional mean temperature
jc' = x - L/2, x being measured from the face T = temperature at the x position
COLD — è
" i t o T T —n HOT
(ip O^otTemp ' r r - j - jp — n
If rotation of any plate section is prevented, then a bending stress of 2a^r7L must be applied, where
a b = - A ¡ E a ( T - T 0)x'dx' -LI2
If both extension and rotation are prevented, the a t stress that must be applied is equal to
a, = -E a(T - To) = EaG(T0 - Tf)
where there is restraint in two directions, Gm, o^, and <5t are multiplied by l/(l-v ), v being Poisson’s ratio and o t = a/ + + o m. In a fully restrained stainless steel element, yielding is produced by a temperature change of 40 K; in structural steels
it is about 100 K. For this reason the plastic components of strain are usually present. Considering a thin cylinder subject to cyclic external quenching when the intermit tent shock quenching is followed by slow rewarming, the distributions of stresses and strain are shown in Fig. 4.2.
