3.1 Solid elastic moduli

One of our interests, as discussed in the introduction, is in porous materials like cement-based materials and in particular gypsum plaster. Therefore, we chose to use the measured elastic moduli of gypsum crystals for the solid backbone moduli. If the backbone moduli tensor is isotropic, then the Young's modulus E of the porous composite will scale with the magnitude of E_s. The shape of the E/E_s vs. porosity graph, for an isotropic backbone, will not depend at all on the value of _s in 2-D [8], and only mildly so in 3-D [2].

However, since gypsum plaster is the material that we are looking to model, we have to deal with the gypsum crystal elastic moduli tensor, which is not isotropic. The gypsum crystal has monoclinic symmetry, so that its 3-D elastic moduli tensor contains 13 independent constants. The measurement of these constants using an acoustic method has been published [13]. As the crystal anisotropy would be very difficult to handle computationally when the crystals are overlapping, and also because we are ultimately interested in real porous materials, which are usually isotropic, we will make the assumption of an isotropic tensor for the solid backbone. An angular average, using eq. ( 11) with the full gypsum elastic moduli tensor, leads to [14]:

The corresponding values of the Young's modulus E and the Poisson's ratio are:

Equations (19) and (20) are the isotropic average of the 3-D tensor. If we want to study 2-D models, a 2-D equivalent must be found. As was discussed in section 2.1, there are two choices for making 2-D moduli from the 3-D moduli: plane strain or plane stress. The plane strain values of the Youngs modulus and Poisson ratio are: E_s = 50 GPa, _s = 0.45. For all the 2-D studies, the plane strain values of the 2-D moduli were used. The plane stress values were used in the data to be presented in Fig. 12 in Section 5.