Semi-infinite-media assumption

Sun and Chen (1988) derived an analytical equation to calculate the heat exchange (Qjj) between colliding spheres based on the weU-established semi-infinite-media assumption. The equation is given by Qy = CQy p, where... 

Assumptions Carbon penetrates into a very thin layer beneath the surface of the component, and thus the component can be modeled as a semi-infinite medium regardless of its thickness or shape. 

The penetration theory in its simplest form represents the case of transient molecular diffusion into a semi-infinite medium It can be applied to real situations if hydrodynamic conditions exist for which such an assumption is approximately valid This would be the case if flow close to the interface is laminar, concentration profiles there are practically nonaal to the interface and time of contact of the phases is reasonably short ... 

We first consider diffusion in a semi-infinite medium and justify this assumption by ensuring later that element A does not reach the end of the bar of B. 

Note that neither initial nor boundary conditions have been applied yet. The above equation is the general solution for infinite and semi-infinite diffusion medium obtained from Boltzmann transformation. The parameters a and b can be determined by initial and boundary conditions as long as initial and boundary conditions are consistent with the assumption that C depends only on q (or ). Readers who are not familiar with the error function and related functions are encouraged to study Appendix 2 to gain a basic understanding. 

In 1885 Joseph Boussinesq (6), trying to extend the validity of these results to the case of axi-symetrical rigid convex punches indenting a flat semi-infinite elastic medium, demonstrates that, without an adequate boundary comlition, the size of the contact area is generally unknown. In ordo to overcome this difficulty, he imposes that normal stresses vanish on the border of the contact area. In other words, the profile of the distorted medium must be tangent to the surfiice of the punch on the border of the contact area. Note that this condition is the same as the condition presupposed by the Hertz s theory. With this assumption, the size oh of the contact area and the penetration depth 5h are completely defined (Figure 1). 

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