The steady-state thermal stress field in an isotropic sphere with temperature dependent properties

Conclusion

For many purposes, the infinite series can be neglected in equations (29) and (30) for the stress field in a hollow sphere and in equations (47) and (48) for a solid sphere. The error estimate from equation (42) or equation (46) may be used as a criterion for deciding whether or not this is permissible. Finally, the method outlined in preceeding discussion is applicable to any one-dimensional thermoelastic problem with only very mild requirements on the variation of Young's Modulus as specified by equation (39). As would be suspected, the use of the procedure for higher dimensional problems entails the specific assumption that second order variations in Young's Modulus are negligibly small if any degree of calculational simplicity is to be retained. Still, this does not require small variations of first order.