Russian Journal of Earth Sciences Russian Journal of Earth Sciences Russian Journal of Earth Sciences 1681-1208 46882 ОРИГИНАЛЬНЫЕ СТАТЬИ ORIGINAL ARTICLES ОРИГИНАЛЬНЫЕ СТАТЬИ Plastic stress field reconstruction based on stress orientations data Plastic stress field reconstruction based on stress orientations data Haderka P Haderka P Galybin A N Galybin A N Escad Design GmbH, Donauw¨orth, Germany ru Escad Design GmbH, Donauw¨orth, Germany ru Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, Moscow, Russia ru Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, Moscow, Russia ru 12 4 1 15 10 11 2021 https://rjes.ru/en/nauka/article/46882/view

This paper presents an investigation of the applicability of the stress trajectories concept and the stress trajectoriesndash;slip lines alternations method to geomechanical problems. We extend our approach introduced for the stress analysis of two-dimensional plastic bodies to the problem of the stress reconstruction in plastic regions of the lithosphere. The method is developed for the Cauchy boundary value problem and utilises the data on principal directions as one of the boundary conditions. For this purpose the first order stress indicators of the World stress map WSM project database release 2008 are utilised in computations. The set of considered boundary conditions is supplemented by the normal derivatives of the stress orientations. Complete formulation of the problem involves a yield condition. Although the general approach is not limited to a specific yield criterion, present calculations are performed for the Mohr-Coulomb criterion. Applications of the method include the stress reconstructions in three regions of the Earth's crust Swiss Alps, Tibetan plateau and Eastern Anatolia. The continuous boundary conditions are derived by an averaging method applied to the discrete data in immediate vicinity of the starting boundary. Thereafter, for the chosen strength parameters of the Mohr-Coulomb theory friction angle and cohesion, the unique grids of stress trajectories and slip lines are determined. These fields are further compared against the WSM data available inside the regions. The computations are made for different strength parameters in order to provide the best fit to the data. The results of the analysis are presented as two plane fields: the map of normalised mean stresses and the grid of corresponding trajectories of principal directions. The normalisation parameter is unknown it represents an initial value of the mean stress in a single node of the boundary, which is a consequence of non-uniqueness of the stress reconstruction problem based on the data on stress orientations alone. The reconstructed stress orientations are compared with the observations from the WSM database.

This paper presents an investigation of the applicability of the stress trajectories concept and the stress trajectoriesndash;slip lines alternations method to geomechanical problems. We extend our approach introduced for the stress analysis of two-dimensional plastic bodies to the problem of the stress reconstruction in plastic regions of the lithosphere. The method is developed for the Cauchy boundary value problem and utilises the data on principal directions as one of the boundary conditions. For this purpose the first order stress indicators of the World stress map WSM project database release 2008 are utilised in computations. The set of considered boundary conditions is supplemented by the normal derivatives of the stress orientations. Complete formulation of the problem involves a yield condition. Although the general approach is not limited to a specific yield criterion, present calculations are performed for the Mohr-Coulomb criterion. Applications of the method include the stress reconstructions in three regions of the Earth's crust Swiss Alps, Tibetan plateau and Eastern Anatolia. The continuous boundary conditions are derived by an averaging method applied to the discrete data in immediate vicinity of the starting boundary. Thereafter, for the chosen strength parameters of the Mohr-Coulomb theory friction angle and cohesion, the unique grids of stress trajectories and slip lines are determined. These fields are further compared against the WSM data available inside the regions. The computations are made for different strength parameters in order to provide the best fit to the data. The results of the analysis are presented as two plane fields: the map of normalised mean stresses and the grid of corresponding trajectories of principal directions. The normalisation parameter is unknown it represents an initial value of the mean stress in a single node of the boundary, which is a consequence of non-uniqueness of the stress reconstruction problem based on the data on stress orientations alone. The reconstructed stress orientations are compared with the observations from the WSM database.

 Numerical approximations and analysis geomechanics plasticity mechanics theory and modelling rheology: crust and lithosphere Numerical approximations and analysis geomechanics plasticity mechanics theory and modelling rheology: crust and lithosphere

Ahorner, Present-day stress field and seismotectonics block movements along major fault zones in western Europe, Tectonophysics, v. 29,, 1975., doi: 10.1016/0040-19517590148-1 Ahorner, Present-day stress field and seismotectonics block movements along major fault zones in western Europe, Tectonophysics, v. 29,, 1975., doi: 10.1016/0040-19517590148-1 Andeweg, Local stress fields and intraplate deformation of Iberia: variations in spatial and temporal interplay of regional stress sources, Tectonophysics, v. 305,, 1999., doi: 10.1016/S0040-19519900004-9 Andeweg, Local stress fields and intraplate deformation of Iberia: variations in spatial and temporal interplay of regional stress sources, Tectonophysics, v. 305,, 1999., doi: 10.1016/S0040-19519900004-9 Becker, Constraints on fault slip rates of the southern California plate boundary from GPS velocity and stress inversions, Geophys. J. Int., 2005., doi: 10.1111/j.1365-246X.2004.02528.x Becker, Constraints on fault slip rates of the southern California plate boundary from GPS velocity and stress inversions, Geophys. J. Int., 2005., doi: 10.1111/j.1365-246X.2004.02528.x Bird, New finite element technique for modelling deformation histories of continents with stratified temperature-dependent rheology, J. Geophys. Res., v. 94, 1989. Bird, New finite element technique for modelling deformation histories of continents with stratified temperature-dependent rheology, J. Geophys. Res., v. 94, 1989. Bird, Thin-plate and thin-shell finite-element programs for forward dynamic modeling of plate deformation and faulting, Computers #x26; Geosciences, v. 25, , 1999., doi: 10.1029/JB094iB04p03967 Bird, Thin-plate and thin-shell finite-element programs for forward dynamic modeling of plate deformation and faulting, Computers #x26; Geosciences, v. 25, , 1999., doi: 10.1029/JB094iB04p03967 Chamlagain, Contemporary tectonic stress field in the Himalaya-Tibet orogen: a view from 2D finite element modeling, J. of Nepal Geological Soc., v. 36, 2007. Chamlagain, Contemporary tectonic stress field in the Himalaya-Tibet orogen: a view from 2D finite element modeling, J. of Nepal Geological Soc., v. 36, 2007. Chang, Interaction between regional stress state and faults: Complementary analysis of borehole in situ stress and earthquake focal mechanism in southeastern Korea, Tectonophysic, 2010., doi: 10.1016/j.tecto.2009.12.012 Chang, Interaction between regional stress state and faults: Complementary analysis of borehole in situ stress and earthquake focal mechanism in southeastern Korea, Tectonophysic, 2010., doi: 10.1016/j.tecto.2009.12.012 Cloetingh, Regional stress field of the Indian plate, Geophys. Res. Lett., v. 12,, 1985., doi: 10.1029/GL012i002p00077 Cloetingh, Regional stress field of the Indian plate, Geophys. Res. Lett., v. 12,, 1985., doi: 10.1029/GL012i002p00077 Coblentz, Tectonic stresses in the African plate: Constraints on the ambient lithospheric stress state, Geology, v. 22, 1994., doi: 10.1130/0091-76131994022#x3C;0831:TSITAP#x3E;2.3.CO;2 Coblentz, Tectonic stresses in the African plate: Constraints on the ambient lithospheric stress state, Geology, v. 22, 1994., doi: 10.1130/0091-76131994022#x3C;0831:TSITAP#x3E;2.3.CO;2 Coblentz, The origins of the intraplate stress field in continental Australia, Earth Planet. Sci. Lett., v. 133,, 1995., doi: 10.1016/0012-821X9500084-P Coblentz, The origins of the intraplate stress field in continental Australia, Earth Planet. Sci. Lett., v. 133,, 1995., doi: 10.1016/0012-821X9500084-P Cox, Axially symmetric plastic deformations in soils, Proc. R. Soc. A: London, v. 254, 1961. Cox, Axially symmetric plastic deformations in soils, Proc. R. Soc. A: London, v. 254, 1961. DeMets, Current plate motions, Geophys. J. Int., v. 101, 1990., doi: 10.1111/j.1365-246X.1990.tb06579.x DeMets, Current plate motions, Geophys. J. Int., v. 101, 1990., doi: 10.1111/j.1365-246X.1990.tb06579.x DeMets, Motion between the Indian, Capricorn and Somalian plates since 20 Ma: implications for the timing and magnitude of distributed lithospheric deformation in the equatorial Indian Ocean, Geophys. J. Int., v. 161, 2005., doi: 10.1111/j.1365-246X.2005.02598.x DeMets, Motion between the Indian, Capricorn and Somalian plates since 20 Ma: implications for the timing and magnitude of distributed lithospheric deformation in the equatorial Indian Ocean, Geophys. J. Int., v. 161, 2005., doi: 10.1111/j.1365-246X.2005.02598.x Dwivedi, Numerical modelling of present-day stress field and deformation pattern in Anatolia, Bull. Fac. Sci., Univ. Ryukyus, v. 89, 2010. Dwivedi, Numerical modelling of present-day stress field and deformation pattern in Anatolia, Bull. Fac. Sci., Univ. Ryukyus, v. 89, 2010. England, Active Deformation of Asia: From kinematics to dynamics, Science, v. 278, 1997., doi: 10.1126/science.278.5338.647 England, Active Deformation of Asia: From kinematics to dynamics, Science, v. 278, 1997., doi: 10.1126/science.278.5338.647 England, Late Quaternary to decadal velocity fields in Asia, J. of Geophys. Res., v. 110, 2005., doi: 10.1029/2004JB003541 England, Late Quaternary to decadal velocity fields in Asia, J. of Geophys. Res., v. 110, 2005., doi: 10.1029/2004JB003541 Flesch, Dynamics of the India--Eurasia collision zone, J. Geophys. Res., v. 106, 2001., doi: 10.1029/2001JB000208 Flesch, Dynamics of the India--Eurasia collision zone, J. Geophys. Res., v. 106, 2001., doi: 10.1029/2001JB000208 Flesch, The dynamics of western North America: Stress magnitudes and the relative role of gravitational potential energy, plate interaction at the boundary, and basal tractions, Geophys. J. Int., v. 169, 2007., doi: 10.1111/j.1365-246X.2007.03274.x Flesch, The dynamics of western North America: Stress magnitudes and the relative role of gravitational potential energy, plate interaction at the boundary, and basal tractions, Geophys. J. Int., v. 169, 2007., doi: 10.1111/j.1365-246X.2007.03274.x Fossen, Structural Geology,, 2010., doi: 10.1017/CBO9780511777806 Fossen, Structural Geology,, 2010., doi: 10.1017/CBO9780511777806 Frocht, Photoelasticity, Vol. 1,, 1941. Frocht, Photoelasticity, Vol. 1,, 1941. Galybin, Determination of elastic stresses from discrete data on stress orientations, Int. J. of Sol. and Struc., v. 41, 2004., doi: 10.1016/j.ijsolstr.2004.04.007 Galybin, Determination of elastic stresses from discrete data on stress orientations, Int. J. of Sol. and Struc., v. 41, 2004., doi: 10.1016/j.ijsolstr.2004.04.007 Ghosh, Contribution of gravitational potential energy differences to the global stress field, Geophys. J. Int., 2009., doi: 10.1111/j.1365-246X.2009.04326.x Ghosh, Contribution of gravitational potential energy differences to the global stress field, Geophys. J. Int., 2009., doi: 10.1111/j.1365-246X.2009.04326.x Gordon, The Antarctic connection, Nature, v. 404,, 2000., doi: 10.1038/35004703 Gordon, The Antarctic connection, Nature, v. 404,, 2000., doi: 10.1038/35004703 G\"olke, Origins of the European regional stress field, Tectonophysics, v. 266, 1996., doi: 10.1016/S0040-19519600180-1 G\"olke, Origins of the European regional stress field, Tectonophysics, v. 266, 1996., doi: 10.1016/S0040-19519600180-1 Gr\"unthal, The recent crustal stress field in central Europe: trajectories and finite element modeling, J. Geophys. Res., v. B97-11, 1992. Gr\"unthal, The recent crustal stress field in central Europe: trajectories and finite element modeling, J. Geophys. Res., v. B97-11, 1992. Haderka, The stress trajectories method for plane plastic problems, Int. J. of Sol. and Struc., v. 48,, 2011., doi: 10.1016/j.ijsolstr.2010.10.016 Haderka, The stress trajectories method for plane plastic problems, Int. J. of Sol. and Struc., v. 48,, 2011., doi: 10.1016/j.ijsolstr.2010.10.016 Heidbach, The World Stress Map database release 2008,, 2008., doi: 10.1594/GFZ.WSM.Rel2008 Heidbach, The World Stress Map database release 2008,, 2008., doi: 10.1594/GFZ.WSM.Rel2008 Hieronymus, A dynamical model for generating Eurasian lithospheric stress and strain rate fields: Effect of rheology and cratons, J. of Geophy. Res., 2008., doi: 10.1029/2007JB004953 Hieronymus, A dynamical model for generating Eurasian lithospheric stress and strain rate fields: Effect of rheology and cratons, J. of Geophy. Res., 2008., doi: 10.1029/2007JB004953 Hill, The Mathematical Theory of Plasticity,, 1950. Hill, The Mathematical Theory of Plasticity,, 1950. Humphreys, North American dynamics and western U.S. tectonics, Rev. Geophys., 2007., doi: 10.1029/2005RG000181 Humphreys, North American dynamics and western U.S. tectonics, Rev. Geophys., 2007., doi: 10.1029/2005RG000181 Keskin, Eastern Anatolia: a hot spot in a collision zone without a mantle plume, The Geological Society of America, Special Paper, 2007., doi: 10.1130/2007.243032 Keskin, Eastern Anatolia: a hot spot in a collision zone without a mantle plume, The Geological Society of America, Special Paper, 2007., doi: 10.1130/2007.243032 Kreemer, An integrated global model of present-day plate motions and plate boundary deformation, Geophys. J. Int., v. 154,, 2003., doi: 10.1046/j.1365-246X.2003.01917.x Kreemer, An integrated global model of present-day plate motions and plate boundary deformation, Geophys. J. Int., v. 154,, 2003., doi: 10.1046/j.1365-246X.2003.01917.x Kuske, Photoelastic stress analysis,, 1974. Kuske, Photoelastic stress analysis,, 1974. Lund, Calculating horizontal stress orientations with full or partial knowledge of the tectonic stress tensor, Geophys. J. Int., 2007., doi: 10.1111/j.1365-246X.2007.03468.x Lund, Calculating horizontal stress orientations with full or partial knowledge of the tectonic stress tensor, Geophys. J. Int., 2007., doi: 10.1111/j.1365-246X.2007.03468.x Martin, Exact bearing capacity calculations using the method of characteristics, Proc. 11th Int. Conf. of IACMAG: Turin, 4,, 2005. Martin, Exact bearing capacity calculations using the method of characteristics, Proc. 11th Int. Conf. of IACMAG: Turin, 4,, 2005. McClusky, GPS constraints of Africa Nubia and Arabia plate motions, Geophys. J. Int., v. 155, , 2003., doi: 10.1046/j.1365-246X.2003.02023.x McClusky, GPS constraints of Africa Nubia and Arabia plate motions, Geophys. J. Int., v. 155, , 2003., doi: 10.1046/j.1365-246X.2003.02023.x Molnar, Relation of the tectonics of eastern China to the India-Eurasia collision: Application of slip-line field theory to large-scale continental tectonics, J. of Geology, v. 5, 1977., doi: 10.1130/0091-761319775#x3C;212:ROTTOE#x3E;2.0.CO;2 Molnar, Relation of the tectonics of eastern China to the India-Eurasia collision: Application of slip-line field theory to large-scale continental tectonics, J. of Geology, v. 5, 1977., doi: 10.1130/0091-761319775#x3C;212:ROTTOE#x3E;2.0.CO;2 Mukhamediev, Theoretical modelling of elastic paleostress fields from in situ indicators, Izvestiya, Physics of the Solid Earth, v. 41, 2005. Mukhamediev, Theoretical modelling of elastic paleostress fields from in situ indicators, Izvestiya, Physics of the Solid Earth, v. 41, 2005. Muller, Regional Patterns of Tectonic Stress in Europe, J. of Geophys. Res., v. 97-B8,, 1992., doi: 10.1029/91JB01096 Muller, Regional Patterns of Tectonic Stress in Europe, J. of Geophys. Res., v. 97-B8,, 1992., doi: 10.1029/91JB01096 Muller, Short-scale variations of tectonic regimes in the western European stress province north of the Alps and Pyrenees, Tectonophysics, v. 275, 1997., doi: 10.1016/S0040-19519700021-8 Muller, Short-scale variations of tectonic regimes in the western European stress province north of the Alps and Pyrenees, Tectonophysics, v. 275, 1997., doi: 10.1016/S0040-19519700021-8 Pauselli, Rheology of the Northern Apennines: Lateral variations of lithospheric strength, Tectonophysics, v. 484, 2010., doi: 10.1016/j.tecto.2009.08.029 Pauselli, Rheology of the Northern Apennines: Lateral variations of lithospheric strength, Tectonophysics, v. 484, 2010., doi: 10.1016/j.tecto.2009.08.029 Regenauer-Lieb, Dilatant plasticity applied to Alpine collision: Ductile void growth in the intraplate area beneath the Eifel volcanic field, J. of Geodyn., v. 27, 1999., doi: 10.1016/S0264-37079700024-0 Regenauer-Lieb, Dilatant plasticity applied to Alpine collision: Ductile void growth in the intraplate area beneath the Eifel volcanic field, J. of Geodyn., v. 27, 1999., doi: 10.1016/S0264-37079700024-0 Sokolovskii, Statics of Granular Media,, 1965. Sokolovskii, Statics of Granular Media,, 1965. Sonder, Western United States extension: How the West was widened, Ann. Rev. Earth Planet. Sci., v. 27, 1999., doi: 10.1146/annurev.earth.27.1.417 Sonder, Western United States extension: How the West was widened, Ann. Rev. Earth Planet. Sci., v. 27, 1999., doi: 10.1146/annurev.earth.27.1.417 Stein, Change in failure stress on the southern San Andreas fault system caused by the Magnitude=7.4 Landers earthquake, Science, v. 258, 1992., doi: 10.1126/science.258.5086.1328 Stein, Change in failure stress on the southern San Andreas fault system caused by the Magnitude=7.4 Landers earthquake, Science, v. 258, 1992., doi: 10.1126/science.258.5086.1328 Steinberger, Largescale lithospheric stress field and topography induced by global mantle circulation, Earth Planet. Sci. Lett., v. 186, 2001., doi: 10.1016/S0012-821X0100229-1 Steinberger, Largescale lithospheric stress field and topography induced by global mantle circulation, Earth Planet. Sci. Lett., v. 186, 2001., doi: 10.1016/S0012-821X0100229-1 Sutherland, Antarctica: A keystone in a changing world, Cooper A. K. et al., eds., Proceedings of the 10th International Symposium on Antarctic Earth Sciences, 2008. Sutherland, Antarctica: A keystone in a changing world, Cooper A. K. et al., eds., Proceedings of the 10th International Symposium on Antarctic Earth Sciences, 2008. Thatcher, Microplate model for the present-day deformation of Tibet, J. Geophys. Res., 2007., doi: 10.1029/2005JB004244 Thatcher, Microplate model for the present-day deformation of Tibet, J. Geophys. Res., 2007., doi: 10.1029/2005JB004244 Vergnolle, Dynamics of continental deformation in Asia, J. Geophys. Res., 2007., doi: 10.1029/2006JB004807 Vergnolle, Dynamics of continental deformation in Asia, J. Geophys. Res., 2007., doi: 10.1029/2006JB004807 Yilmaz, Kinematics of the East Anatolian Fault Zone between Turkoglu Kahramanmaras and Celikhan Adiyaman, eastern Turkey, Earth Planets Space, v. 58, 2006. Yilmaz, Kinematics of the East Anatolian Fault Zone between Turkoglu Kahramanmaras and Celikhan Adiyaman, eastern Turkey, Earth Planets Space, v. 58, 2006. Zoback, Tectonic stress field of North America and relative plate motions, Neotectonics of North America, Slemmons, D. B., Engdahl, E. R., Zoback, M. D., Blackwell, D. D. eds.,, 1991. Zoback, Tectonic stress field of North America and relative plate motions, Neotectonics of North America, Slemmons, D. B., Engdahl, E. R., Zoback, M. D., Blackwell, D. D. eds.,, 1991. Zoback, Steady-State Failure Equilibrium and Deformation of Intraplate Lithosphere, International Geology Review, v. 44, 2002., doi: 10.2747/0020-6814.44.5.383 Zoback, Steady-State Failure Equilibrium and Deformation of Intraplate Lithosphere, International Geology Review, v. 44, 2002., doi: 10.2747/0020-6814.44.5.383 Zoback, Lithosphere Stress and Deformation, Earthquake Seismology -- Treatise on Geophysics,, v. 6, 2007. Zoback, Lithosphere Stress and Deformation, Earthquake Seismology -- Treatise on Geophysics,, v. 6, 2007.