Assessment of Tensorial and Scalar Damage Models for an Isotropic Thermally Cracked Rock Under Confining Pressure Using Experimental Data: Continuum Damage Mechanics Versus Effective Medium Theory

Abstract

Many models of damage or cracking of isotropic solids consider a single damage/crack density variable. Based on both continuum damage mechanics (CDM) and effective medium theory (EMT), we model the impact of isotropic damage in the form of microcracks on the elastic properties of an isotropic solid. For each approach, we consider the complete tensorial description of the elastic moduli involving two damage/crack density variables $D_{1}$ and $D_{2}$, or $\alpha$ and $\beta$; and possible scalar approximations involving a single variable (with either $D_{2}$ = 0, or $\beta$ = 0). We assess the accuracy of scalar approximations commonly employed for each approach against laboratory measurements of ultrasonic wave velocities and density obtained on a dry and isotropic specimen of thermally cracked Carrara Marble (CM) subjected to an increasing confining pressure up to 50 MPa. Overall, this laboratory dataset and the CDM and EMT modelling and inversion results reported here suggest that: (i) irreversible thermal cracking and microcrack opening occur after heating and sudden cooling of the CM specimen, whereas reversible and progressive microcracks’ closure occurs with increasing confining pressure; (ii) tensorial damage/cracking models involving either two damage variables (CDM) or two crack density variables (EMT) fit equally well and virtually perfectly the laboratory data for any confining pressure tested; (iii) single scalar approximation models commonly used in CDM or the EMT models give comparable results to their complete tensorial counterparts, which is particularly true for the CDM approach; (iv) single scalar approximations derived from the CDM approach, and assuming a constant Poisson’s ratio of the cracked rock, reproduces all the elastic moduli more accurately than the corresponding scalar approximation derived from the EMT approach, where a constant ratio of Young’s modulus to Poisson’s ratio is assumed instead; and (v) it is more reliable to use a tensorial rather than a scalar description of the effect of reversible microcrack closure with pressure on all elastic parameters, including Poisson’s ratio. If the impact of reversible microcrack closure is accounted for, then a single scalar description of irreversible thermal damage with the CDM approach is remarkably accurate.