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properties profiles in FGM


I am studying the mechanical response (behavior) of an FGM ( Functionally Graded Material )material that has a varying properties in the x direction according to the following Young's modulus formula 

   E(x)=E0*exp(ß*x)  for: 0<x<L

with: ß the non-homogeneity parameter of the material 

changing density for FGM in umat

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Hello friends,
Here is the Abaqus user subroutine for Functionally Graded Material with direct integration method. This Umat is based on S.H.Song's Umat. Obviously the coefficient AAA in calculating E is the ratio of young moduli of  two sides of the plate. With this method, one can change young modulus through the thickness.

But I have problem with changing density through the thickness, meaning from density of ceramic to density of metal.
I would be grateful if anyone can help me. Any help would be appreciated.

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