A Statistical Damage Constitutive Model for Geomaterials under Plane-Strain Biaxial Stress State

Abstract

The mechanical behavior of geomaterials under plane-strain biaxial stress state (PSBSS, a special case of biaxial stress state) is often considered in geotechnical structures such as highwall and longwall coal pillars. In this study, a modified statistical damage constitutive model based on Weibull distribution was established to explain the mechanical behavior of rocks under the PSBSS. The modified Wiebols-Cook criterion, Drucker-Prager criterion, and extremum method were adopted in this model to estimate the peak strength of rock, the strength level of microscopic element, and the statistical parameters of model, respectively. Besides, laboratory tests for brittle and ductile geomaterials under PSBSS were conducted using the modified surface instability detection apparatus to validate the accuracy of the proposed statistical damage model. Finally, the relationships between mechanical parameters and statistical parameters were studied and discussed.

Constitutive model of rock triaxial damage based on the rock strength statistics

“…In addition, some experts have proposed a lot of distribution functions of the meso-element strength, including Weibull distribution, normal distribution, lognormal distribution and maximum entropy distribution, etc. (Chen and Xu et al, 2018;Li et al, 2015;Li et al, 2017;Liu et al, 2017;Xu and Yang, 2017;Zhang et al, 2017;Zhou et al, 2017;Yang 2016;Yao et al, 2019;Zhang et al, 2015). Therefore, we assumed that the distribution of meso-element strength obeys a lognormal distribution and its probability density distribution function f p ð Þ is given by…”

“…Let N D represent the number of damaged meso-elements and N represent the total number of meso-elements. Therefore, the damage variable D can be defined as (Li et al, 2012(Li et al, , 2015Liu et al, 2017;Zhang et al, 2017).…”

“…Finally, considering a fact that internal defects in the rock distributes randomly, Zhou et al (2017) used statistical methods to establish statistical damage constitutive models based on continuum damage mechanics. In addition, other researchers (Bruning et al, 2019;Chen and Xu et al, 2018;Ismael and Konietzky, 2019;Li et al, 2015Li et al, , 2017Shen et al, 2019;Zhang et al, 2017;Li &Jimenez, 2018;Li et al,2019;Tiwari & Latha, 2019;Wang & Cai, 2019) also established their statistical damage constitutive models to describe mechanical behaviors of rocks by considering that the rock mesoscopic element (meso-element) strength obeys the random distribution. Therefore, the introduction of statistical damage theory provides a reliable reference for the development of rock damage constitutive theory.…”

“…represents the action area of undamaged parts; and r 00 1 represents the stress of damaged parts.Assuming that damage variable D ¼ AÀA$A(Zhang et al, 2017), we can obtain from equation(17)…”

Statistical Distributions for Damage Modelling: A Retrospect

“…constitutive fracture damage model for fissured rock mass is established based on the deformation characteristics of microcracks under compression. By employing the basic theory of damage mechanics inZhang, Wan, Wang, Ma, Zhang, and Cheng (2017), they established a statistical damage constitutive model for rocks under the PSBSS (Plane-Strain Biaxial Stress State). The constitutive model of rocks subjected to cyclic stress-temperature was proposed inZhou, Xia, Zhao, Mei, and Zhou (2017) and model for cemented sand considering the residual strength and initial compaction phase was discussed inTan, Yuan, Shi, Zhou, and Li (2018).…”