Mechanical response of elastic materials with density dependent Young modulus

The experimental as well as theoretical engineering literature on porous structures such as metal foams, aerogels or bones often relies on the standard linearised elasticity theory, and, simultaneously, it frequently introduces the concept of “density dependent Young modulus”. We interpret the concept of “density dependent Young modulus” literally, that is we consider the linearised elasticity theory with the generalised Young modulus being a function of the current density, and we briefly summarise the existing literature on theoretical justification of such models. Subsequently we numerically study the response of elastic materials with the “density dependent Young modulus” in several complex geometrical settings. In particular, we study the extension of a right circular cylinder, the deflection of a thin plate, the bending of a beam, and the compression of a cube subject to a surface load, and we quantify the impact of the density dependent Young modulus on the mechanical response in the given setting. In some geometrical settings the impact is almost nonexisting—the results based on the classical theory with the constant Young modulus are nearly identical to the results obtained for the density dependent Young modulus. However, in some cases such as the deflection of a thin plate, the results obtained with constant/density dependent Young modulus differ considerably despite the fact that in both cases the infinitesimal strain condition is well satisfied.