Capturing Elastic Properties and Their Dependencies as a Precursor to Understanding Damage
Authors: Knippel E. P. , Villaquiran Vargas A. P. , Xiong Q. , Hampton J. C. 
Published: September 2023

ABSTRACT
Experimental rock mechanics testing provides an effective method for measuring physical properties, their dependencies, and their evolution due to the addition of localized microcracks in a controlled laboratory environment. In order to understand the contributions of microcrack parameters to first order changes in compliance, the behavior of initial undamaged properties of a material must be thoroughly understood as a function of stress, load path, and load history. To that end, we perform a comprehensive study of elastic properties and their dependence on testing parameters for a variety of materials exhibiting linear, nonlinear, and varying levels of anisotropic elastic stiffnesses. We utilize a programmatically controlled triaxial testing apparatus to perturb specimens instrumented with strain gauges and ultrasonic transducers. Elastic moduli are then measured along multiple discrete loading paths as a function of stress, with the upper limit calculated through an experimental derived failure envelope. Once the experimental results of those effects are isolated, we will then be able to assess the accuracy of limiting the modeled damage related changes in compliance to solely crack density, length, and orientation in a homogenized medium.

INTRODUCTION
The relationship between stress and strain is a constitutive law governed by properties intrinsic to a material. This is often assumed to be linearly elastic, where strain increases constantly with stress and deformation fully recovers once the applied stress is released. This is not always the case and is readily observed in experimental data, however the relationship is close enough for undamaged materials that the assumption is still valid in a variety of applications (Jaeger et al. (2007)). As stress and strain are both second order tensors, a fourth order tensor is required to relate these properties. This compliance tensor contains 81 components, however due to symmetry this can be reduced to 36 similarly to how both stress and strain contain six independent coefficients out of the total nine. For layered and isotropic materials, the number of independent coefficients can further be reduced. The compliance tensor for layered or vertically transversely isotropic (VTI) materials contain eight independent coefficients. Materials that are isotropic in all directions contain three independent coefficients. These coefficients can be calculated through experimental testing, where applied stress is manipulated along various load paths and resulting strain measured. There exists dynamic elastic moduli as well, measured through the use of ultrasonic transducers, where high frequency and low strain amplitude ultrasonic waves propagate through the material. The differences between static (low frequency and higher strain amplitude) and dynamic (high frequency and low strain amplitude) stress-strain perturbations often result in subtle differences between the moduli, though typical relationships between dynamic and static elastic moduli have been studied extensively and is out of the scope of this paper. VTI samples with non-uniform static elastic moduli are expected to have similar differences in velocity and dynamic moduli.