Non-linear elastic behaviour of damaged rocks

Abstract

S U M M A R YThe pervasive damage of rocks by microcracks and voids stron ly affects their macroscopic elastic properties. To evaluate the damage effects, w d derive here the macroscopic stress-strain relations for a 3-D elastic solid with nohinteracting cracks embedded inside a homogeneous matrix. The cracks considered are oriented either perpendicular to the maximum tension axis, or perpendicular to the maximum compression axis. In the first case they dilate during loading and in the second case they contract during loading. We derive a solution for the elastic energy of this rock following the self-consistent scheme of Budiansky & O'Connell ( 1976). The solution describes the stress-strain relations in terms of Ad and pd, which are the modified Lame constants, and an additional parameter y. The latter accounts for the non-linear behaviour of the solid and is related to crack density. The solution predicts a non-linear elastic rheology even for an infinitesimal strain of ~<0.001, and abrupt change in the elastic moduli when the loading reverses from uniaxial compression to uniaxial tension.We use the derived solution to analyse rock-mechanics experiments in which beams of Indiana limestone were deformed under four-point loading. This configuration provides simultaneously the apparent tensile and compressive moduli for small strains. The apparent moduli fit the effective elastic moduli calculated with the present damage model well, including the differences between tensile and compressive moduli.

Structural Properties and Deformation Patterns of Evolving Strike-slip Faults: Numerical Simulations Incorporating Damage Rheology

“…Appendix 2 presents an overview of the variable-force boundary conditions, and Appendix 3 applies the conditions in a test study verifying the viscoelastic component of our code. Additional details on the employed damage model and comparisons of results with laboratory fracture and friction data are given by LYAKHOVSKY et al (1997aLYAKHOVSKY et al ( ,b, 2005, HAMIEL et al (2004HAMIEL et al ( , 2006, .…”

“…A final analytical constraint for healing parameters can be derived from the convexity condition for macroscopic failure used in our damage rheology framework (LYAKHOVSKY et al, 1997a). This stability condition indicates that near the surface, where normal stress is low compared to shear stress and the strain invariants ratio is approximately zero, the maximum sustainable (stable) damage level is approximately a = 0.75.…”

“…The parameter n 0 is a yielding threshold separating states of deformation involving material degradation (da/dt > 0) when n > n 0 , and material healing (da/dt < 0) when n < n 0 . LYAKHOVSKY (1995) andLYAKHOVSKY et al (1997a) related this parameter to the angle of internal friction by considering the critical shear stress for Mohr-Coulomb sliding. They obtained n 0 = -0.8 for rock with internal friction coefficient of f = 0.6 and Poisson's ratio m = 0.25, and noted that this value varies only slightly (-0.7 to -0.9) for rocks with Poisson's ratio between 0.2 and 0.3.…”

“…166, 2009 Damage Rheology Models of Strike-slip Fault Zone Evolutiondamage. LYAKHOVSKY et al (1997a) showed that the leading term of the damage evolution equation, satisfying energy conservation and nonnegative entropy production, can be written as…”

“…The first two terms of equation (1) give the classical strain potential of linear elasticity (e.g., MALVERN, 1969). The third term may be derived using the effective medium theory of BUDIANSKY and O'CONNELL (1976) for non-interacting cracks that dilate and contract in response to tension and compression (LYAKHOVSKY et al, 1997b), or by expanding the strain energy potential as a general second-order function of I 1 and I 2 and eliminating non-physical terms . The kinetic aspects of the damage rheology involve making the elastic moduli functions of the damage state variable, and deriving an equation for the evolution of Vol.…”