skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Examination of Mean Stress Calculation Approaches in Rock Mechanics

Abstract

The mean stress, as a fundamental statistical property of a group of stress data, is essential for stress variability characterisation. However, currently in rock mechanics, the mean stress is customarily and erroneously calculated by separately averaging the principal stress magnitudes and orientations. In order to draw the attention of our community to the appropriate approach for stress variability characterisation, here we compare the customary scalar/vector mean with that obtained by the mathematically rigorous tensorial approach—the Euclidean mean. Calculation of mean stress using both a small group of actual in situ stress measurement results and a large group of simulated stress data (obtained using the combined finite– discrete element method, FEMDEM) demonstrates that the two approaches yield different results. Further investigation of these results shows that the scalar/vector approach may yield non-unique and non-orthogonal mean principal stresses, and these may deviate significantly from those of the Euclidean mean. Our calculations and comparisons reveal that the scalar/ vector approach is deficient because it processes the principal stress magnitudes and orientations separately as independent quantities and ignores the connection between them. Conversely, the tensorial approach appropriately averages the tensors that simultaneously carry not only the information of stress magnitudes and orientations, but alsomore » the inherent relations between them. Therefore, arbitrarily using scalar/vector mean stress of in situ stress measurements as input in further rock engineering analyses may yield significantly erroneous results. Finally, we advise that stress data should be statistically processed in a tensorial manner using tensors referred to a common Cartesian coordinate system.« less

Authors:
ORCiD logo [1];  [2]
  1. Univ. of Toronto, ON (Canada). Dept. of Civil Engineering; Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  2. Univ. of Toronto, ON (Canada). Dept. of Civil Engineering
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE; Natural Sciences and Engineering Research Council of Canada (NSERC)
OSTI Identifier:
1479961
Report Number(s):
[LA-UR-18-25280]
[Journal ID: ISSN 0723-2632]
Grant/Contract Number:  
[AC52-06NA25396; 491006]
Resource Type:
Accepted Manuscript
Journal Name:
Rock Mechanics and Rock Engineering
Additional Journal Information:
[ Journal Volume: 52; Journal Issue: 1]; Journal ID: ISSN 0723-2632
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
58 GEOSCIENCES; Stress tensor; Rock mass; Mean stress; Euclidean mean; FEMDEM

Citation Formats

Gao, Ke, and Harrison, John P. Examination of Mean Stress Calculation Approaches in Rock Mechanics. United States: N. p., 2018. Web. doi:10.1007/s00603-018-1568-0.
Gao, Ke, & Harrison, John P. Examination of Mean Stress Calculation Approaches in Rock Mechanics. United States. doi:10.1007/s00603-018-1568-0.
Gao, Ke, and Harrison, John P. Mon . "Examination of Mean Stress Calculation Approaches in Rock Mechanics". United States. doi:10.1007/s00603-018-1568-0. https://www.osti.gov/servlets/purl/1479961.
@article{osti_1479961,
title = {Examination of Mean Stress Calculation Approaches in Rock Mechanics},
author = {Gao, Ke and Harrison, John P.},
abstractNote = {The mean stress, as a fundamental statistical property of a group of stress data, is essential for stress variability characterisation. However, currently in rock mechanics, the mean stress is customarily and erroneously calculated by separately averaging the principal stress magnitudes and orientations. In order to draw the attention of our community to the appropriate approach for stress variability characterisation, here we compare the customary scalar/vector mean with that obtained by the mathematically rigorous tensorial approach—the Euclidean mean. Calculation of mean stress using both a small group of actual in situ stress measurement results and a large group of simulated stress data (obtained using the combined finite– discrete element method, FEMDEM) demonstrates that the two approaches yield different results. Further investigation of these results shows that the scalar/vector approach may yield non-unique and non-orthogonal mean principal stresses, and these may deviate significantly from those of the Euclidean mean. Our calculations and comparisons reveal that the scalar/ vector approach is deficient because it processes the principal stress magnitudes and orientations separately as independent quantities and ignores the connection between them. Conversely, the tensorial approach appropriately averages the tensors that simultaneously carry not only the information of stress magnitudes and orientations, but also the inherent relations between them. Therefore, arbitrarily using scalar/vector mean stress of in situ stress measurements as input in further rock engineering analyses may yield significantly erroneous results. Finally, we advise that stress data should be statistically processed in a tensorial manner using tensors referred to a common Cartesian coordinate system.},
doi = {10.1007/s00603-018-1568-0},
journal = {Rock Mechanics and Rock Engineering},
number = [1],
volume = [52],
place = {United States},
year = {2018},
month = {9}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 2 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

Combined single and smeared crack model in combined finite-discrete element analysis
journal, January 1999


The Statistical Analysis of Orthogonal Orientation Data
journal, May 1989

  • Lisle, Richard J.
  • The Journal of Geology, Vol. 97, Issue 3
  • DOI: 10.1086/629309

Generation of random stress tensors
journal, April 2017


Influence of Large Syncline on In Situ Stress Field: A Case Study of the Kaiping Coalfield, China
journal, July 2016


Multivariate distribution model for stress variability characterisation
journal, February 2018


Influence of boundary constraints on stress heterogeneity modelling
journal, July 2018


The stress state of the Earth’s lithosphere: Results of statistical processing of the world stress-map data
journal, February 2013

  • Koptev, A. I.; Ershov, A. V.; Malovichko, E. A.
  • Moscow University Geology Bulletin, Vol. 68, Issue 1
  • DOI: 10.3103/S0145875213010067

Finite strain, finite rotation quadratic tetrahedral element for the combined finite-discrete element method
journal, August 2009

  • Xiang, Jiansheng; Munjiza, Antonio; Latham, John-Paul
  • International Journal for Numerical Methods in Engineering, Vol. 79, Issue 8
  • DOI: 10.1002/nme.2599

Characterizing in situ stress domains at the AECL Underground Research Laboratory
journal, October 1990

  • Martin, C. D.
  • Canadian Geotechnical Journal, Vol. 27, Issue 5
  • DOI: 10.1139/t90-077

Modelling stress-dependent permeability in fractured rock including effects of propagating and bending fractures
journal, January 2013

  • Latham, John-Paul; Xiang, Jiansheng; Belayneh, Mandefro
  • International Journal of Rock Mechanics and Mining Sciences, Vol. 57
  • DOI: 10.1016/j.ijrmms.2012.08.002

Effects of geomechanical changes on the validity of a discrete fracture network representation of a realistic two-dimensional fractured rock
journal, September 2014

  • Lei, Qinghua; Latham, John-Paul; Xiang, Jiansheng
  • International Journal of Rock Mechanics and Mining Sciences, Vol. 70
  • DOI: 10.1016/j.ijrmms.2014.06.001

In situ strength and failure mechanisms of migmatitic gneiss and pegmatitic granite at the nuclear waste disposal site in Olkiluoto, Western Finland
journal, October 2015


Overcoring in highly stressed granite— the influence of microcracking
journal, January 1991

  • Martin, C. D.; Christiansson, R.
  • International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 28, Issue 1
  • DOI: 10.1016/0148-9062(91)93233-V

Mean and dispersion of stress tensors using Euclidean and Riemannian approaches
journal, May 2016


Trends in relationships between measured in-situ stresses and depth
journal, August 1978

  • Brown, E. T.; Hoek, E.
  • International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 15, Issue 4
  • DOI: 10.1016/0148-9062(78)91227-5

Scalar-valued measures of stress dispersion
journal, June 2018


Overcoring in highly stressed granite: Comparison of USBM and modified CSIR devices
journal, January 1991

  • Martin, C. D.; Christiansson, R. C.
  • Rock Mechanics and Rock Engineering, Vol. 24, Issue 4
  • DOI: 10.1007/BF01045032

In Situ rock stresses and their measurement in the U.K.—Part I. The current state of knowledge
journal, December 1988

  • Hudson, J. A.; Cooling, C. M.
  • International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 25, Issue 6
  • DOI: 10.1016/0148-9062(88)90976-X

The state of stress in the upper part of the earth's crust
journal, September 1969


Confidence intervals for In Situ stress measurements
journal, April 1990

  • Walker, J. R.; Martin, C. D.; Dzik, E. J.
  • International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 27, Issue 2
  • DOI: 10.1016/0148-9062(90)94864-P

Investigating the Relationship Between Far-Field Stress and Local Values of the Stress Tensor
journal, January 2017


On the validation of DEM and FEM/DEM models in 2D and 3D
journal, August 2009

  • Xiang, Jiansheng; Munjiza, Antonio; Latham, John‐Paul
  • Engineering Computations, Vol. 26, Issue 6
  • DOI: 10.1108/02644400910975469

Comparison of deterministic with stochastic fracture models in water-flooding numerical simulations
journal, November 2009

  • Belayneh, Mandefro W.; Matthai, Stephan K.; Blunt, Martin J.
  • AAPG Bulletin, Vol. 93, Issue 11
  • DOI: 10.1306/07220909031

In-situ stress measurements and regional stress field assessment of the Beishan area, China
journal, August 2013


Stress, instability and design of underground excavations
journal, October 2003

  • Martin, C. D.; Kaiser, P. K.; Christiansson, R.
  • International Journal of Rock Mechanics and Mining Sciences, Vol. 40, Issue 7-8
  • DOI: 10.1016/S1365-1609(03)00110-2

Correlation Between Fracture Network Properties and Stress Variability in Geological Media
journal, May 2018


Confidence intervals for in situ stress measurements
journal, December 1994

  • Jupe, A. J.
  • International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 31, Issue 6
  • DOI: 10.1016/0148-9062(94)90013-2

Updating the use of the CCBO cell in Japan: overcoring case studies
journal, October 2003


    Works referencing / citing this record:

    Uncertainty in In Situ Stress Estimations: A Statistical Simulation to Study the Effect of Numbers of Stress Measurements
    journal, June 2019

    • Feng, Yu; Harrison, John P.; Bozorgzadeh, Nezam
    • Rock Mechanics and Rock Engineering, Vol. 52, Issue 12
    • DOI: 10.1007/s00603-019-01891-9