Abstract:
In this paper, we consider a non-Euclidean continuum model for which the structure of defects in the material is characterized by an internal metric and scalar curvature. It is shown that the irrotational displacement field for points of this medium is composed of elastic displacements (in the absence of defects) and the field which characterizes the deviation of the internal geometry of the model from Euclidean geometry. The corresponding components of the internal stresses are the sum of elastic stresses and the self-equilibrated stresses determined by the scalar curvature. The exact solution for the vortex field of dislocations is constructed, and conditions of the existence of a nonzero stress field parametrized by a scalar curvature in the absence of external forces are formulated.
Citation:
M. A. Guzev, “Structure of kinematic and force fields in the Riemannian continuum model”, Prikl. Mekh. Tekh. Fiz., 52:5 (2011), 39–48; J. Appl. Mech. Tech. Phys., 52:5 (2011), 709–716
\Bibitem{Guz11}
\by M.~A.~Guzev
\paper Structure of kinematic and force fields in the Riemannian continuum model
\jour Prikl. Mekh. Tekh. Fiz.
\yr 2011
\vol 52
\issue 5
\pages 39--48
\mathnet{http://mi.mathnet.ru/pmtf1520}
\elib{https://elibrary.ru/item.asp?id=17255698}
\transl
\jour J. Appl. Mech. Tech. Phys.
\yr 2011
\vol 52
\issue 5
\pages 709--716
\crossref{https://doi.org/10.1134/S002189441105004X}
Linking options:
https://www.mathnet.ru/eng/pmtf1520
https://www.mathnet.ru/eng/pmtf/v52/i5/p39
This publication is cited in the following 18 articles:
Oxana V. Sadovskaya, Vladimir M. Sadovskii, “Mathematical modelling of fracture waves in a blocky medium with thin compliant interlayers”, Phil. Trans. R. Soc. A., 382:2277 (2024)
Cong Zhang, Zhende Zhu, Shanyong Wang, Chong Shi, Wangyang Li, “Seismic response and deformation mechanism of near-fault deep tunnels in a strong earthquake area”, Acta Geotech., 18:9 (2023), 4847
Mikhail Guzev, Advanced Structured Materials, 155, Advanced Materials Modelling for Mechanical, Medical and Biological Applications, 2022, 213
Cong Zhang, Zhende Zhu, Shanyong Wang, Xuhua Ren, Chong Shi, “Stress wave propagation and incompatible deformation mechanisms in rock discontinuity interfaces in deep‐buried tunnels”, Deep Underground Science and Engineering, 1:1 (2022), 25
Yingji Bao, Binsong Jiang, Tongguang Ni, “Theory and Numerical Simulation of Deep Rock Mass Based on a Non-Euclidean Model”, Scientific Programming, 2022 (2022), 1
M. A. Guzev, W. Liu, Ch. Qi, E. P. Riabokon, “Compensating role self-balanced stress fields in constructing nonsingular solutions using a non-Euclidean model of a continuous medium for an incompressible sphere”, J. Appl. Mech. Tech. Phys., 62:5 (2021), 736–741
Mikhail A. Guzev, Modeling in Geotechnical Engineering, 2021, 61
Mikhail A. Guzev, Smart Innovation, Systems and Technologies, 214, Smart Modelling For Engineering Systems, 2021, 75
Wei Liu, Mikhail Guzev, Chengzhi Qi, “Non-Euclidean model for description of residual stresses in planar deformations”, Applied Mathematical Modelling, 90 (2021), 615
Yingji Bao, Binsong Jiang, “Incompatible Deformation Model of Rocks with Defects around a Thick-Walled Cylinder”, Processes, 9:12 (2021), 2215
Y.D. Shou, X.P. Zhou, Q.H. Qian, “A critical condition of the zonal disintegration in deep rock masses: Strain energy density approach”, Theoretical and Applied Fracture Mechanics, 97 (2018), 322
Y. D. Shou, X. P. Zhou, Q. H. Qian, “Dynamic Model of the Zonal Disintegration of Rock Surrounding a Deep Spherical Cavity”, Int. J. Geomech., 17:6 (2017)
Jing Bi, Xiao Ping Zhou, “Numerical Simulation of Zonal Disintegration of the Surrounding Rock Masses Around a Deep Circular Tunnel Under Dynamic Unloading”, Int. J. Comput. Methods, 12:03 (2015), 1550020
Xiaoping Zhou, Qihu Qian, Hanfei Song, “The effects of three-dimensional penny-shaped cracks on zonal disintegration of the surrounding rock masses around a deep circular tunnel”, Acta Mechanica Solida Sinica, 28:6 (2015), 722
Mikhail A. Guzev, “Non-Classical Solutions of Critical Rock Behavior”, AMR, 891-892 (2014), 1663
Mikhail Guzev, Chengzhi Qi, Jiping Bai, Kairui Li, “Equilibrium equations and boundary conditions of strain gradient theory in arbitrary curvilinear coordinates”, Journal of the Mechanical Behavior of Materials, 23:5-6 (2014), 169
Mikhail A. Guzev, “Non-classical solutions of a continuum model for rock descriptions”, Journal of Rock Mechanics and Geotechnical Engineering, 6:3 (2014), 180
Xiao-Ping Zhou, Yun-Dong Shou, “Excavation-induced zonal disintegration of the surrounding rock around a deep circular tunnel considering unloading effect”, International Journal of Rock Mechanics and Mining Sciences, 64 (2013), 246
Citation in format AMSBIB
Contact us: