  In diffusion processes of solid materials, such as in the classical thermal
shock problem and the recent lithium ion battery, the maximum diffusion induced
stress (DIS) is a very important quantity. However a widely accepted, accurate
and easy-to-use quantitative formula on it still lacks. In this paper, by
normalizing the governing equations, an almost analytical model is developed,
except a single-variable function of the dimensionless Biot number which cannot
be determined analytically and is then given by a curve. Formulae for various
typical geometries and working conditions are presented. If the stress and the
diffusion process are fully coupled (i.e. stress-dependent diffusion), as in
lithium ion diffusion, the normalized maximum DIS can be characterized by a
two-variable function of a dimensionless coupling parameter and the Biot
number, which is obtained numerically and presented in contour plots. Moreover,
it is interesting to note that these two parameters, within a wide range, can
be further approximately combined into a single dimensionless parameter to
characterize the maximum DIS. These formulae together with curves/contours
provide engineers and materialists a simple and easy way to quickly obtain the
stress and verify the reliability of materials under various typical diffusion
conditions. Via energy balance analysis, the model of diffusion induced
fracture is also developed. It interestingly predicts that the spacing of
diffusion induced cracks is constant, independent of the thickness of specimen
and the concentration difference. Our thermal shock experiments on alumina
plates validate these qualitative and quantitative theoretical predictions,
such as the constant crack spacing and the predicted critical temperature
difference at which the cracks initiate. Furthermore, the proposed model can
interpret the observed hierarchical crack patterns.
