There are many models:

Phong or Blinn variant: Are very simple, can work fine for glossy or not so rough materials. The original were empirical and not physically correct, but there is a variant that makes them physically correct. These models don't work for rough materials and mirrors and are unable to produce the effect that is the subject of this thread. For plastics combined with Fresnel can work pretty well.

Schlick: Is the same thing as Phong or Blinn, but require less computation time. For very specular materials produce the same thing as Phong, but when the material becomes rough the behavior of Sclick is very poor, Phong is much better.

Cook-Sparrow-Torrance: Physically correct for rough and little glossy surfaces. Normally are used for metals. Fail to work with glossy and mirror like surfaces.

He-Torrance: The most complete physically correct model. It can deal with rough to mirror-like materials and polarization effects. It is derived from Beckmann-Spizichino model and is very computational expensive. As I know no rendering engine has implemented this model.

Lambert: The most simple and ideal model for diffuse illumination. Pure Lambertian objects do not exist in Nature.

Lommel-Seeliger: The other fundamental model for difusse illumination. Very few rendering engines has this model.

Oren-Nayar: Model of difusse illumination for very rough materials. It works fine with ceranics and many diffuse materials.

Hapke: Model for dusty surfaces, a must have for rendering planets, moon and vegetation.

And many others.....

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What I am doing is working on the complete model of Kircchoff-Beckmann-Spizichino or He-Sillon-Torrance variant and trying to find an approximation that preserve all its features, but has a small computation time.

 

Stupidity also evolves!