Effects of porosity and microcracking on elastic constants

As indicated earlier, the upper and lower bounds for the elastic constants become wider as the modulus ratio of the constituents increases. Thus, the bounding approaches become much less useful. This is particularly true when the second phase is porosity. In this ease, the lower bounds are zero and there is, therefore, great difficulty in accurately estimating the elastic constants of porous materials with this approach. The upper bounds in Eqs. (3.12) and (3.14) can still be used by setting the bulk and shear moduli of the second phase to zero. Porosity will always lead to a decrease in elastic modulus as not only is the load-bearing area of a material being reduced by the pores but also the stress becomes concentrated near the pores (see Section 4.9). 

The experimental data are values extrapolated to zero porosity. 

Exact solutions exist for bodies containing a dilute suspension of pores. For example, for a body containing a low concentration of spherical pores, the MacKenzie (1950) solution for the Young s modulus E of the porous body is given by 

The next level of refinement in developing constitutive relationships for porous bodies is to assume a pore shape. Thus, the self-consistent solutions can also be 

The idea that the elastic moduli of ceramic powder compacts are close to zero has also led to the idea that the change in the elastic constants is simply given by the degree of densification. Thus, it has been suggested that 

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