Modeling of Plastic Deformation Based on the Theory of an Orthotropic Cosserat Continuum
Tóm tắt
In the paper, the plastic deformation
of heterogeneous materials is analyzed by direct numerical
simulation based on the theory of an elastic-plastic orthotropic
Cosserat continuum, with the plasticity condition taking into
account both the shear and rotational mode of irreversible
deformation. With the assumption of a block structure of a
material with elastic blocks interacting through compliant
plastic interlayers, this condition imposes constraints on the
shear components of the asymmetric stress tensor, which
characterize shear, and on the couple stresses, which
irreversibly change the curvature characteristics of the
deformed state of the continuum upon reaching critical values.
The equations of translational and rotational motion together
with the governing equations of the model are formulated as a
variational inequality, which correctly describes both the state
of elastic-plastic deformation under applied load and the state
of elastic unloading. The numerical implementation of the
mathematical model is performed using a parallel computing
algorithm and an original software for cluster multiprocessor
systems. The developed approach is applied to solve the problem
of compressing a rectangular brick-patterned blocky rock mass by
a rough nondeformable plate rotating with constant acceleration.
The effect of the yield stress of the compliant interlayers on
the stress-strain state of the rock mass in shear and bending is
studied. The field of plastic energy dissipation in the rock
mass is analyzed along with the fields of displacements,
stresses, couple stresses, and rotation angle of structural
elements. The obtained results can help to validate the
hypothesis about the predominant effect of curvature on plastic
strain localization at the mesolevel in microstructural
materials.
Tài liệu tham khảo
Panin, V.E., Fomin, V.M., and Titov, V.M., Physical Principles of Mesomechanics of Surface Layers and Internal Interfaces in a Solid under Deformation, Phys. Mesomech., 2003, vol. 6, no. 3, pp. 5–14.
Panin, V.E., Panin, A.V., and Moiseenko, D.D., Physical Mesomechanics of a Deformed Solid as a Multilevel System. II. Chessboard-Like Mesoeffect of the Interface in Heterogeneous Media in External Fields, Phys. Mesomech., 2007, vol. 10, no. 1–2, pp. 5–14.
Panin, V.E., Egorushkin, V.E., and Panin, A.V., Nonlinear Wave Processes in a Deformable Solid as in a Multiscale Hierarchically Organized System, Phys. Usp., 2012, vol. 55, pp. 1260–1267.
Panin, V.E., Likhachev, V.A., and Grinyaev, Yu.V., Structural Levels of Deformation in Solids, Novosibirsk: Nauka, 1985.
Guzev, M.A. and Makarov, V.V., Deformation and Fracture of Highly Compressed Rocks around Underground Workings, Vladivostok: Dalnauka, 2007.
Rattez, H., Stefanou, I., Sulem, J., Veveakis, M., and Poulet, T., Numerical Analysis of Strain Localization in Rocks with Thermo-Hydro-Mechanical Couplings Using Cosserat Continuum, Rock Mech. Rock Eng., 2018, vol. 51, no. 10, pp. 3295–3311. https://doi.org/10.1007/s00603-018-1529-7
Tarasov, B.G., Hitherto Unknown Shear Rupture Mechanism as a Source of Instability in Intact Hard Rocks at Highly Confined Compression, Tectonophysics, 2014, vol. 621, pp. 69–84. https://doi.org/10.1016/j.tecto.2014.02.004
Tarasov, B.G., Guzev, M.A., Sadovskii, V.M., and Cassidy, M.J., Modelling the Mechanical Structure of Extreme Shear Ruptures with Friction Approaching Zero Generated in Brittle Materials, Int. J. Fracture, 2017, vol. 207, no. 1, pp. 87–97. https://doi.org/10.1007/s10704-017-0223-14
Cosserat, E. et Cosserat, F., Théorie des Corps Déformables, Chwolson’s Traité Physique, Paris: Librairie Scientifique A. Hermann et Fils, 1909, pp. 953–1173.
Günther, W., Zur Statik und Kinematik des Cosseratschen Kontinuums, Abhandlungen der Braunscheigschen Wissenschaftlichen Gesellschaft, Braunschweig: F. Vieweg and Sohn, 1958, vol. 10, pp. 195–213.
Sadovskii, V.M. and Sadovskaya, O.V., Modeling of Elastic Waves in a Blocky Medium Based on Equations of the Cosserat Continuum, Wave Motion, 2015, vol. 52, pp. 138–150. https://doi.org/10.1016/j.wavemoti.2014.09.008
Friedrichs, K.O., Symmetric Hyperbolic Linear Differential Equations, Commun. Pure Appl. Math., 1954, vol. 7, no. 2, pp. 345–392. https://doi.org/10.1002/cpa.3160070206
Godunov, S.K., Equations of Mathematical Physics, Moscow: Nauka, 1979.
Godunov, S.K. and Romenskii, E.I., Elements of Continuum Mechanics and Conservation Laws, New York: Kluwer Academic/Plenum Publishers, 2003. https://doi.org/10.1007/978-1-4757-5117-8
Sadovskii, V.M., Discontinuous Solutions in Dynamic Elastoplastic Problems, Moscow: Nauka, 1997.
Sadovskaya, O. and Sadovskii, V., Mathematical Modeling in Mechanics of Granular Materials, Ser.: Advanced Structured Materials, vol. 21, Heidelberg: Springer, 2012. https://doi.org/10.1007/978-3-642-29053-4
Sadovskii, V.M., Sadovskaya, O.V., and Varygina, M.P., RF Patent No. 2012614823, Software for Solving Two-Dimensional Dynamic Problems of the Moment Theory of Elasticity (2Dyn_Cosserat), Moscow: FIPS, 2012, no. 3(80).
Tyumentsev, A.N., Ditenberg, I.A., Korotaev, A.D., and Denisov, K.I., Lattice Curvature Evolution in Metal Materials on Meso- and Nanostructural Scales of Plastic Deformation, Phys. Mesomech., 2013, vol. 16, no. 4, pp. 319–334.