A cross-anisotropic elastoplastic model applied to sedimentary rocks

Bayesian inference of rock strength anisotropy: Uncertainty analysis of the Hoek–Brown failure criterion

“…Yet, this isotropic strength criterion cannot characterize strength anisotropy in rocks. We follow the elastoplastic models of Mánica et al 71 and Forero et al 70 and formulate an anisotropic strength criterion with the help of the unitless scalar orientation parameters, 𝐶 N and 𝐶 S . As the strength of anisotropic rocks depends on how the loading is applied relative to the orientation of their anisotropy planes, this approach provides a simple way to relate the loading direction to the anisotropy direction.…”

“…The formulation as proposed in this paper is relatively simple to implement and has an additional advantage since our approach does not use complex components of elastoplastic models as in previous studies. 70,71 Algorithm 1 Anisotropic HB strength criterion with scaled stress tensor Compute the stress rotation matrix, σ r , using Equation (7)…”

“…74 For example, in geotechnical engineering, there is a steadily growing body of literature on the application of Bayesian analysis to inverse modeling and quantification of model parameter and output uncertainty. 51,[55][56][57][58]60,70 Bayesian inference allows for an exact description of parameter uncertainty by treating the parameters (and nuisance variables) as probabilistic variables with joint posterior probability density function, 𝑝(θ|σ 1 ). This multivariate distribution, the so-called posterior parameter distribution, is the consequence of two antecedents, a prior distribution, 𝑝(θ), which captures our initial degree of beliefs in the values of the model parameters, θ, and a likelihood function, 𝐿(θ|σ 1 ), which quantifies, by the rules of probability theory, the level of confidence (= conditional belief) in the parameter values in light of the observed peak strength data, σ1 = (σ 1,1 , σ1,2 , … , σ1,𝑛 ), where the subscripts of σ1,2 refer to sample 2 of the peak strength σ1 , etc.…”

“…We follow Ref. 70 and use the scaling factors, 𝐶 N and 𝐶 S , to link the orientation of the planes of anisotropy with the loading direction. Our HB model formulation assumes isotropic parameters for intact rock, which can be measured directly from rock samples using triaxial tests.…”

A micromechanical hydro-mechanical-damage coupled model for layered rocks considering multi-scale structures

“…In consideration of the oriented macroscopic discontinuities, many anisotropic elastoplastic constitutive models have been proposed by treating the layered rocks as transversely isotropic media. The elastoplastic models are mainly obtained from the extension of the isotropic formulations by introducing transversely isotropic Hooke's law for elasticity, orientation dependence of anisotropic failure criterion for plasticity (Chen et al 2012, Ismael et al 2019, Wang et al 2019, Forero et al 2020, and directional-dependent Biot coefficients for poroelasticity (Cheng 1997, Guayacan-Carrillo 2017. In the work done by Guayacan-Carrillo et al (2017), for instance, a fully coupled anisotropic poroelastic model was developed with the accounting of transversely isotropic elastic stiffness tensor, two different Biot coefficients in the direction parallel and perpendicular to the plane of isotropy.…”