Self-consistency between input and output data for models describing elastic properties of fractured media: does conventional models satisfy this criterion?

Models that adequately describe the effect of crack-like defects on the elastic moduli of solids are one of the key "ingredients" needed to obtain diagnostic conclusions about the structural features of the material. The change in the velocities of longitudinal and shear elastic waves depending on the pressure is one of the most popular methods of measuring the connection of these moduli with the cracks present in the material. For commonly considered models with an isotropic crack orientation (which makes the medium on average isotropic), the measurement of these two velocities is sufficient to determine two independent moduli (for example, the shear and bulk moduli) through which other characteristics of interest can be expressed. In this case, the applied pressure, gradually closing the cracks, is a control parameter that regulates the concentration of cracks.

It is quite natural when constructing models to obtain expressions relating the elastic moduli with the crack concentration (the latter cannot be to directly monitored when the applied pressure is varied). In this regard, some additional considerations are used about the relationship of crack concentration to pressure, which allows one to relate the model expressions with the moduli measured during the pressure variation. Assuming some approximations relating the pressure and concentration with free fitting parameters in the model, it is possible to achieve the best agreement of model with the experimental dependences on pressure. This approach looks natural and is conventionally used, resulting in apparently satisfactory agreement between the model predictions and the measurement data.

Here we show that this apparent agreement is often achieved at the expense of strong violation of self-consistency between the input data fed into the model and the output of the model. This violation is far from obvious in conventional approaches based on the use of an auxiliary (and not directly measurable) relationship between concentration and pressure. To find out the fact of either violation or fulfillment of the condition of self-consistency, here we describe such a form of the model, in which its input parameters can be expressed in terms of experimentally measured values (in contrast to the crack concentration that cannot be monitored as a function of pressure). In the proposed description of the fractured material, the cracks are characterized by the shear- and normal compliances, the ratio of which is not a priori fixed and can be extracted from the experimental data.The proposed procedure of interpretation of experimental pressure dependences allows one to explicitly verify the model self-consistency and assess the elastic properties of real cracks that is many cases appear to be strongly different from the properties intrinsic to the standard penny-shape-crack model.

The reported study was supported by RFBR, project number 19-05-00536.