# Talk:Hashin–Shtrikman bounds

## Reader feedback: A tutorial of generating HS ...

The HS upper bound is obtained for an example with 2 constituents by substituting the larger value of bulk modulus into K2 as and the smaller value as K1. The lower bound is obtained by swapping K1 and K2. Porosity is the variable (between 0 and 1)

Conceptually speaking, you can imagine the case of porosity approaching zero decreasing the sorting of a bunch of concentric spheres. Smaller spheres fill in the open spaces and add to the effective stiffness. The effective stiffness of the material obviously depends if the higher bulk modulus is on the inside, or outside of each concentric sphere.

Considering the diagram as an example, the upper bound is achieved when K2 = 36.6 GPa (quartz), and K1 = 2.2 GPa (brine). The lower bound is achieved when K2 = 2.2 GPa (brine), and K1 = 36.6 GPa (quartz).

As, for generating bounds on effective elastic properties for more than 2 phases, that is a more difficult problem. You may need to approximate two phases as belonging to one. In other words, break the problem down into two 2 phase problems. With that said, these bounds will likely be violated for more than two phases, and for real cases where grains aren't perfect concentric spheres. More complicated mixing problems may need numerical modelling, but HS is a good place to start when you are modelling elastic properties, say of a grain covered by cement.