Abstract
An analytical solution of the one-dimensional problem of a spherical cavity expanding at a constant velocity from a point in a half-space occupied by a plastic medium has been obtained. Impact compressibility of the medium is described using linear Hugoniot’s adiabat. Plastic deformation obeys the Mohr-Coulomb yield criterion with constraints on the value of maximum tangential stresses according to Tresca’s criterion. In the assumption of rigid-plastic deformation (the elastic precursor being neglected), incompressibility behind the shockwave front and the equality of the propagation velocities of the fronts of the plastic wave and the plane shockwave defined by linear Hugoniot’s adiabat, a boundary-value problem for a system of two first-order ordinary differential equations for the dimensionless velocity and stress depending on the self-similar variable is formulated. A closed-form solution of this problem has been obtained in the form of a stationary running wave—a plastic shockwave propagating in an unperturbed half-space. The solution is a generalization of the earlier obtained analytical solution for a medium with the Mohr-Coulomb plasticity condition. A formula for determining a critical pressure (a minimal pressure required for the nucleation of a cavity, accounting for internal friction in the framework of Mohr-Coulomb yield criterion), which is a generalization of the earlier solution for an ideal plastic medium with Tresca’s criterion, has been derived. The resulting critical pressure was compared with a numerical solution in a full formulation at cavity propagation velocities close to zero in a wide range of the parameters of the Mohr-Coulomb yield criterion. The approximation inaccuracy of the introduced formula does not exceed 6% for the internal friction coefficient varying over the entire permissible range and the initial value of yield strength changing by three orders of magnitude. The effect of constraining the limiting value of maximal tangential stresses on the distribution of dimensionless stresses behind the shockwave front has been examined. Formulas for determining the range of cavity expansion velocities, within which a simple solution for a medium with Tresca’s plasticity condition is applicable, have been derived. The obtained solution can be used for evaluating resistance to high-velocity penetration of rigid strikers into low-strength soil media.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Abali, B. E., Muller, W., & dell’Isola, F. (2017). Theory and computation of higher gradient elasticity theories based on action principles. Archive of Applied Mechanics, 87, 1495–1510.
Alibert, J. J., Seppecher, P., & dell’Isola, F. (2003). Truss modular beams with deformation energy depending on higher displacement gradients. Mathematics and Mechanics of Solids, 8, 51–73.
Arlery, M., Gardou, M., Fleureau, J. M., & Mariotti, C. (2010). Dynamic behaviour of dry and water-saturated sand under planar shock conditions. The International Journal of Impact Engineering, 37, 1–10. https://doi.org/10.1016/j.ijimpeng.2009.07.009.
Auffray, N., dell’Isola, F., Eremeyev, V., Madeo, A., & Rosi, G. (2013). Analytical continuum mechanics à la Hamilton-Piola: Least action principle for second gradient continua and capillary fluids. Mathematics and Mechanics of Solids.
Balandin, V. V., Bragov, A. M., Igumnov, L. A., Konstantinov, A. Yu., Kotov, V. L., & Lomunov, A. K. (2015). Dynamic deformation of soft soil media: Experimental studies and mathematical modeling. Mechanics of Solids, 50(3), 286–293. https://doi.org/10.3103/S002565441503005X.
Barchiesi, E., Spagnuolo, M., & Placidi, L. (2018). Mechanical metamaterials: A state of the art. Mathematics and Mechanics of Solids.
Bazhenov, V. G., & Kotov, V. L. (2008). Method for identifying elastoplastic properties of ground media by penetration of impactors. Mechanics of Solids, 4(43), 678–686. https://doi.org/10.3103/S002565440804016X.
Bazhenov, V. G., Kotov, V. L., Kochetkov, A. V., Krylov, S. V., & Feldgun, V. R. (2001). On wave processes in soil subjected to a surface charge explosion. Mechanics of Solids, 2(36), 62–68.
Bazhenov, V. G., Bragov, A. M., & Kotov, V. L. (2009). Experimental-theoretical study of the penetration of rigid projectiles and identification of soil properties. Journal of Applied Mechanics and Technical Physics, 6(50), 1011–1019. https://doi.org/10.1007/s10808-009-0135-6.
Ben-Dor, G., Dubinsky, A., & Elperin, T. (2005). Ballistic impact: Recent advances in analytical modeling of plate penetration dynamics–A review. Applied Mechanics Reviews, 58, 355–371.
Bragov, A. M., & Grushevskii, G. M. (1993). Influence of the moisture content and granulometric composition on the shock compressibility of sand. Technical Physics Letters, 19, 385–386.
Bragov, A. M., Lomunov, A. K., Sergeichev, I. V., Proud, W., Tsembelis, K., & Church, P. (2005). A method for determining the main mechanical properties of soft soils at high strain rates (103–105 s–1) and load amplitudes up to several gigapascals. Technical Physics Letters, 31(6), 530–531. https://doi.org/10.1134/1.1969791.
Bragov, A. M., Balandin, V. V., Igumnov, L. A., Kotov, V. L., Krushka, L., & Lomunov, A. K. (2018). Impact and penetration of cylindrical bodies into dry and water-saturated sand. International Journal of Impact Engineering, 122, 197–208.
dell’Isola, F., & Hutter, K. (1998). A qualitative analysis of the dynamics of a sheared and pressurized layer of saturated soil. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 454(1980), 3105–3120.
dell’Isola, F., & Hutter, K. (1999). Variations of porosity in a sheared pressurized layer of saturated soil induced by vertical drainage of water. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 455(1988), 2841–2860.
dell’Isola, F., Guarascio, M., & Hutter, K. (2000). A variational approach for the deformation of a saturated porous solid. A second-gradient theory extending Terzaghi’s effective stress principle. Archive of Applied Mechanics, 70(5), 323–337.
dell’Isola, F., Seppecher, P., & Madeo, A. (2012). How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: Approach “à la D’Alembert”. Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 63(6), 1119–1141.
dell’Isola, F., Andreaus, U., & Placidi, L. (2015). At the origins and in the vanguard of peri-dynamics, non-local and higher gradient continuum mechanics. An underestimated and still topical contribution of Gabrio Piola. Mathematics and Mechanics of Solids, 20, 887–928.
dell’Isola, F., Madeo, A., & Seppecher, P. (2016). Cauchy tetrahedron argument applied to higher contact interactions. Archive for Rational Mechanics and Analysis, 219(3).
dell’Isola, F., Della Corte, A., & Giorgio, I. (2017). Higher-gradient continua: The legacy of Pio-la, Mindlin, Sedov and Toupin and some future research perspectives. Mathematics and Mechanics of Solids, 22, 852–872.
dell’Isola, F., Seppecher, P., Alibert, J. J., Lekszycki, T., Grygoruk, R., Pawlikowski, M., et al. (2019). Pantographic metamaterials: An example of mathematically driven design and of its technological challenges. Continuum Mechanics and Thermodynamics, 31(4), 851–884.
Del Vescovo, D., & Giorgio, I. (2014). Dynamic problems for metamaterials: Review of existing models and ideas for further research. International Journal of Engineering Science, 80, 153–172.
Dianov, M. D., Zlatin, N. A., Mochalov, S. M., et al. (1977). Shock compressibility of dry and water-saturated sand. Soviet Technical Physics Letters, 2, 207–208.
Forrestal, M. J., & Longcope, D. B. (1982). Closed-form solution for forces on conical-nosed penetrators into geological targets with constant shear strength. Mechanics of Materials, 1, 285–295.
Forrestal, M. J., & Luk, V. K. (1988). Dynamic spherical cavity-expansion in a compressible elastic-plastic solid. The Journal of Applied Mechanics, 55(2), 275–279 https://doi.org/10.1115/1.3173672.
Forrestal, M. J., & Luk, V. K. (1992). Penetration into soil targets. The International Journal of Impact Engineering, 12(3), 427–444. https://doi.org/10.1016/0734-743X(92)90167-R.
Grigoryan, S. S. (1960). On basic concepts of soil dynamics. Journal of Applied Mathematics and Mechanics, 24(6), 1604–1627. https://doi.org/10.1016/0021-8928(60)90013-7.
Kong, X. Z., Wu, H., Fang, Q., & Peng, Y. (2017). Rigid and eroding projectile penetration into concrete targets based on an extended dynamic cavity expansion model. The International Journal of Impact Engineering, 100, 13–22.
Kotov, V. L. (2008). Studying the applicability of the auto-modeling analysis of the problem of expansion of a spherical cavity in a compressible medium for determining the pressure on the surface of the striker-soil contact. Problems of Strength and Plasticity, 70, 123–131. (in Russian).
Kotov, V. L. (2019). Approximation of stresses in the vicinity of the cavity, expanding at a constant speed in a medium with the Mohr-Coulomb yield criterion. Problems of Strength and Plasticity, 81(2), 1–10. (in Russian).
Kotov, V. L., Linnik, E. Yu., Tarasova, A. A., & Makarova, A. A. (2011). The analysis of approximate solutions of a spherical cavity expansion in a soil medium problem. Problems of Strength and Plasticity, 73, 58–63. (in Russian).
Kotov, V. L., Balandin, V. V., Bragov, A. M., Linnik, E. Yu., & Balandin, V. V. (2013). Using a local-interaction model to determine the resistance to penetration of projectiles into sandy soil. Journal of Applied Mechanics and Technical Physics, 54(4), 612–621 https://doi.org/10.1134/S0021894413040123.
Lagunov, V. A., & Stepanov, V. A. (1963). Measurements of the dynamic compressibility of sand under high pressures. Zh. Prikl. Mekh. Tekhn. Fiz., 1, 88–96. [J. Appl. Mech. Tech. Phys. (Engl. Transl.)].
Linnik, E. Yu., Kotov, V. L., Tarasova, A. A., & Gonik, E. G. (2012). The solution of the problem of the expansion of a spherical cavity in a soil medium assuming incompressibility beyond the shock front. Problems of Strength and Plasticity., 74, 49–58. (In Russian).
Omidvar, M., Iskander, M., & Bless, S. (2012). Stress-strain behavior of sand at high strain rates. International Journal of Impact Engineering, 49, 192–213. https://doi.org/10.1016/j.ijimpeng.2012.03.004.
Omidvar, M., Iskander, M., & Bless, S. (2014). Response of granular media to rapid penetration. International Journal of Impact Engineering, 66, 60–82. https://doi.org/10.1016/j.ijimpeng.2013.12.004.
Omidvar, M., Malioche, Jeanne D., Bless, S., & Iskander, M. (2015). Phenomenology of rapid projectile penetration into granular soils. International Journal of Impact Engineering, 85, 146–160. https://doi.org/10.1016/j.ijimpeng.2015.06.002.
Placidi, L., Barchiesi, E., Turco, E., & Rizzi, N. L. (2016). A review on 2D models for the description of pantographic fabrics. Zeitschrift für angewandte Mathematik und Physik, 67(5).
Placidi, L., Andreaus, U., & Giorgio, I. (2017). Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model. Journal of Engineering Mathematics.
Rahali, Y., Giorgio, I., Ganghoffer, J.-F., & dell’Isola, F. (2015). Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices. International Journal of Engineering Science, 97.
Rosenberg, Z., & Dekel, E. (2008). A numerical study of the cavity expansion process and its application to long-rod penetration mechanics. International Journal of Impact Engineering, 35(3), 147–154.
Shi, C., Wang, M., Li, J., & Li, M. (2014). A model of depth calculation for projectile penetration into dry sand and comparison with experiments. International Journal of Impact Engineering, 73, 112–122. https://doi.org/10.1016/j.ijimpeng.2014.06.010.
Shi, C., Wang, M., Zhang, K., Cheng, Y., & Zhang, X. (2015). Semi-analytical model for rigid and erosive long rods penetration into sand with consideration of compressibility. International Journal of Impact Engineering, 83, 1–10.
Sun, Q., Sun, Y., Liu, Y., Li, R., & Zhao, Y. (2017). Numerical analysis of the trajectory stability and penetration ability of different lateral-abnormal projectiles for non-normal penetration into soil based on modified integrated force law method. International Journal of Impact Engineering, 103, 159–168.
Veldanov, V. A., Markov, V. A., Pusev, V. I., Ruchko, A. M., Sotskii, M. Yu., & Fedorov, S. V. (2011). Computation of non-deformable striker penetration into low strength obstacles using piezoelectric accelerometry data. Technical Physics, 56(7), 992–1002. https://doi.org/10.1134/S1063784211070231.
Acknowledgements
The work was done under financial support from the Grant of Russian Fund of Fundamental Researches (Project No. 19-08-00430).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Кotov, V.L., Linnik, E.Y., Sabaeva, T.A. (2021). Analyzing the Problem of a Spherical Cavity Expansion in a Medium with Mohr-Coulomb-Tresca’s Plasticity Condition. In: dell'Isola, F., Igumnov, L. (eds) Dynamics, Strength of Materials and Durability in Multiscale Mechanics. Advanced Structured Materials, vol 137. Springer, Cham. https://doi.org/10.1007/978-3-030-53755-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-53755-5_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-53754-8
Online ISBN: 978-3-030-53755-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)