A physically-based elastoplastic damage model for quasi-brittle geomaterials under freeze-thaw cycles and loading

Micro‐Thermomechanical Modeling of Rocks With Temperature‐Dependent Friction and Damage Laws

“…where ⃦ ⃦ s l ⃦ ⃦ is the Euclidean norm of the second-order deviatoric local stress tensor s l = K : σ l ; meanwhile, p l = 1 3 δ : σ l denotes the mean local stress; and μ* symbolizes the friction coefficient of mesocracks within the REV. It is noted that in the established works devoted to constitutive modeling, the friction coefficient μ* is conventionally considered constant for the sake of simplification (Molladavoodi, 2015;Qu et al, 2021;Ren et al, 2022;. Recently, however, a series of investigations have revealed that the friction coefficient is not a constant material parameter; rather, it is dependent on a number of factors, for example, ambient temperature, stress state, moisture, damage state, friction rate, and spatio-temporal scale (Ben-David & Fineberg, 2011;Putelat et al, 2011;Scholz, 2019;Zhang & Ma, 2021;C.…”

“…Z. Zhu et al., 2016; Zhao, Shao, & Zhu, 2018; Zhao, Lv, et al., 2023), we assume that the sliding on mesocracks obeys the Coulomb‐type friction law: $$$\mathcal{F}=\left\Vert {\boldsymbol{s}}^{l}\right\Vert -{\mu }^{\ast }{p}^{l}\le 0,$$$ where $$\left\Vert {\boldsymbol{s}}^{l}\right\Vert $$ is the Euclidean norm of the second‐order deviatoric local stress tensor $${\boldsymbol{s}}^{l}=\mathbb{K}:{\boldsymbol{\sigma }}^{l}$$; meanwhile, $${p}^{l}=\frac{1}{3}\boldsymbol{\delta }:{\boldsymbol{\sigma }}^{l}$$ denotes the mean local stress; and μ * symbolizes the friction coefficient of mesocracks within the REV. It is noted that in the established works devoted to constitutive modeling, the friction coefficient μ * is conventionally considered constant for the sake of simplification (Molladavoodi, 2015; Qu et al., 2021; Ren et al., 2022; Zhao, Shao, & Zhu, 2018). Recently, however, a series of investigations have revealed that the friction coefficient is not a constant material parameter; rather, it is dependent on a number of factors, for example, ambient temperature, stress state, moisture, damage state, friction rate, and spatio‐temporal scale (Ben‐David & Fineberg, 2011; Putelat et al., 2011; Scholz, 2019; Zhang & Ma, 2021; C. X. Zhao, Liu, et al., 2023; Thom et al., 2023).…”