D • CONDUCTION OF HEAT 



where v, E, and /3 represent Poisson's ratio, Young's modulus, and the 

 coefficient of thermal expansion, respectively. 



By the use of Eq. 7-8, the mean dimensionless temperature in the 

 cylinder is 



-.) = ^(j,. L a') / ^ 2.rfr = j^ f^ e{., r,)^ 



e{ 



Accordingly, solving the thermoelastic equations for the heated cylinder 

 [11] we obtain: 



Va = ^^igj^ = ^W - ^(1, r.) (8-2) 



v^ = ^^%^ = ^W - ^(«. ^») (8-3) 



Thus the dimensionless stress on the boundary is numerically equal to the 

 difference between the mean dimensionless temperature and the boundary 

 temperature, with i? ^ indicating compression and tension, respectively. 

 In the course of heating from initially uniform temperature, the bound- 

 ary stresses rise from zero, pass through respective maxima, and subse- 

 quently decrease toward zero as the shell approaches thermal equihbrium 

 at d = 1. These maxima depend only on the Biot and thickness numbers. 

 Analogous expressions for thermal stresses in a cylinder with Newtonian 

 heat flux at the outer radius r = 6 are given in [13]. 



D,9. Remarks on Thermal Shock. A serious difficulty in the use of 

 refractories under rapidly varying temperature conditions is their tend- 

 ency to crack, chip, or spall due to the presence of thermal stresses set up 

 by large spatial variations of the temperature within the material. The 

 phrase "thermal shock" is commonly employed to describe the failure of 

 the material under transient stress conditions associated with large tem- 

 perature gradients. In the literature on the subject, various criteria have 

 been proposed to express the resistance to thermal shock in terms of 

 material properties and size factors. Recently [I4] an attempt has been 

 made to include the effect of heating parameters in the criterion for resist- 

 ance to thermal shock. On this basis it can be shown that previously pro- 

 posed criteria are special cases corresponding to various regimes of the 

 heat transfer coefficient h. 



The physical role of material and heating parameters in the occurrence 

 of thermal shock can be described, assuming that the material behaves 

 elastically, by comparison of transient thermal stresses with maximum 

 stresses such as, say, the yield stress of the material. 



This has been done in [I4] on the model of a slab of thickness 2d 

 heated (or cooled) symmetrically by Newtonian heat transfer through its 



< 270 ) 



