# Victor Berdichevsky

Professor Emeritus, Mechanical Engineering

## Biography

** Publications**

- Recollections of V.I. Yudovich, IUTAM Symposium on Hamiltonian dynamics, vortex structures and turbulence, 2006c.
- Berdichevsky, V.L., Continuum theory of dislocations revisited, Continuum Mech. Thermodyn., v. 18, 195-222, 2006b.
- Berdichevsky, V.L., On thermodynamics of crystal plasticity, Scripta Materialia, v. 54, 711-716, 2006a.
- Berdichevsky, V.L., Averaged equations of ideal fluid turbulence, Continuum Mech. Thermodyn., v.19, 133-175, 2007b.
- Berdichevsky, V.L., Entropy of microstructure, J. Mech. Phys. Solids, v. 56, 3, 742-771, 2008a.
- Berdichevsky, V.L., On statistical mechanics of vortex lines, in: IUTAM Symposium on Hamiltonian dynamics, vortex structures and turbulence, Springer-Verlag, 205-210, 2008b.
- Berdichevsky, V.L., Variational principle in dynamics of vortex filament, Phys. Review E, v. 78, 036304, 2008c.
- Berdichevsky, V.L., On variational features of vortex flows, Continuum Mech. Thermodyn., v. 20, 219-229, 2008d.
- Berdichevsky, V.L., Distribution of minimum values of stochastic functionals, Networks and heterogeneous media, v. 3, 437-460, 2008e.
- Berdichevsky, V.L., Estimates of effective characteristics of random structures in terms of effective characteristic of periodic structures, Int. J. Eng. Sci., v. 47, 165-169, 2009a.
- Berdichevsky, V.L., Continuum theory of dislocations: 40 years later, L.I. Sedov 100-aniversary volume, Moscow, 2009b.
- Berdichevsky, V.L. and L. Shirkov, Comparison of three-dimensional and two-dimensional statistical mechanics of shear layers for flow between two parallel plates, V.I. Yudovich 75-aniversary volume, Izv. Vuzov, 2009c.
- Berdichevsky, V.L., An asymptotic theory of sandwich plates, Int. J. Eng. Sci., v. 48, 3, 383-404, 2010a.
- Berdichevsky, V.L., Nonlinear theory of hard-skin plates and shells, Int. J. Eng. Sci., v. 48, 357-369, 2010b.
- Berdichevsky, V. L., Distributions of local fields in composites, Int. J. Eng. Sci., v. 48, 1006-1019, 2010c.
- Pilipchuk, V.N., Berdichevsky, V.L., Ibrahim, R.A., Thermo-mechanical coupling in cylindrical bending of sandwich plates, Composite Structures, v. 92, 2632-2640, 2010d.
- Berdichevsky, V. L., On dislocation model of grain boundaries, Continuum Mech. Thermodyn., v. 23, 185-209, 2011a.
- Berdichevsky, V.L., On the origin of microstructure self-similarity in severe plastic deformation, J. Mech. Phys. Solids, v. 60, 1-13, 2012a.
- Berdichevsky, V.L., Universal grain size distribution, most chaotic microstructures and tessellation condition, Int. J. Eng. Sci., v. 57, 24-35, 2012b.
- Berdichevsky,V.L., Thermodynamics of microstructure evolution: grain growth, Int. J. Eng. Sci., v. 57, 50-78, 2012c.
- Shirkov L., Berdichevsky,V.L., Comparison of three-dimensional and two-dimensional statistical mechanics of shear layers for flow between two parallel plates, Continuum Mech. Thermodyn., v. 25, 269-272, 2013a.
- Berdichevsky,V.L., Beyond classical thermodynamics: grain growth, Emerging Materials Research, v. 2, 66-70, 2013b.
- Ye, Z., Berdichevsky V.L., Yu, W., An equivalent classical plate model of corrugated structures, Int. J. Solids Struct., v. 51, 2073-2083, 2014a.
- Berdichevsky,V.L., Overcoming paradoxes of Drucker-Prager theory for unconsolidated granular matter, Int. J. Eng. Sci., v. 83, 174-186, 2014b.
- Berdichevsky, V.L., Homogenization in probabilistic terms: the variational principle and some approximate solutions, J. Mech. Phys. Solids, 87, 86-114, 2016a.
- Berdichevsky, V.L., Relations between grain boundary character distribution, orientation distribution and two-point orientation density, Int. J. Eng. Sci., 99, 1-12, 2016b.
- Berdichevsky, V.L., J.N. Herman, On rheology of cross-linked polymers: 1. Slippage of polymer chains and its macroscopic modeling, Int. J. Eng. Sci., 100, 183-198, 2016c.
- Ostberg, D., Berdichevsky, V.L., Electric field fluctuations in conductors with spherical inclusions, Mech. Materials, 93, 81-95, 2016d.
- Berdichevsky, V.L., On a continuum theory of dislocation equilibrium, Int. J. Eng. Sci., 106, 10-28, 2016e.
- Berdichevsky, V.L., Energy of dislocation networks, Int. J. Eng. Sci., 103, 35-44, 2016f.

## Older Publications

**MICROMECHANICS**

- Berdichevsky, V., and Sedov, L., "Dynamic Theory of Continuously Distributed Dislocations: Its Relation to Plasticity Theory," J. Appl. Math. Mech. (PMM), Vol. 31, No. 6, pp. 989-1006, 1967.
- Berdichevsky, V., "The Size of the Edge Dislocation,"Moscow University Mechanics Bulletin, Vol. 27, No. 1, pp. 24-27, 1972.
- Berdichevsky, V., Hazzledine, P., and Shoykhet, B., "Micromechanics of DiffusinalCreep," Int. J. Engng Sci., Vol. 35, No.10/11, pp. 103-1032, 1997.
- Berdichevsky, V., "On Collective Nucleation of Dislocations," Scripta Materialia, Vol. 51, pp. 561-563, 2004.
- "On Failure of Continuum Plasticity Theories on Small Scales," (with D. Dimiduk), Scripta Materialia, Vol. 52, pp. 1017-1019, 2005.
- "Homogenization in Micro-Plasticity," J. Mechanics and Physics of Solids, Vol. 53, pp.2457-2469, 2005.
- "On the Microcrack Nucleation in Brittle Solids," (with K.C. Le), International Journal of Fracture, Vol. 133: L47-L54, 2005.
- "On thermodynamics of Crystal Plasticity," Scripta Materialia, Vol. 54, pp. 711-716,2006.
- "Dislocation Nucleation and Work Hardening in Anti-Plane Shear," (with K.C. Le),Continuum Mechanics and Thermodynamics, Vol. 18, pp. 455-467, 2007.
- Continuum theory of dislocations revisited, Continuum Mechanics and Thermodynamics, 18, 195-222, 2006.
- Entropy of microstructure. Journal of the Mechanics and Physics of Solids, doi:10.1016/j.jmps.2007.07.004, 2007.
- Distribution of minimum values of stochastic Functionals. Networks and heterogeneous media, v. 3, 437-460, 2008
- Estimates of effective characteristics of random structures in terms of effective characteristic of periodic structures. International Journal of Engineering Science,v.47, 165-169, 2009
- Universal grain size distribution, most chaotic microstructures and tessellation condition [Submitted]
- Thermodynamics of microstructure evolution: grain growth [Submitted]

**HOMOGENIZATION**

- Berdichevsky, V., "Spatial Averaging of Periodic Structures," Soviet Physics - Doklady, Vol. 20, No. 5, pp. 334-335, 1975.
- Berdichevsky, V., "On Averaging of Periodic Systems," J. Appl. Math. Mech. (PMM), Vol. 41, No. 6, 1977, pp. 1010-1023.
- Berdichevsky, A. and Berdichevsky, V., "Flow of an Ideal Liquid Past a Periodic System of Bodies," Fluid Dynamics, Vol. 13, pp. 797-810, 1978.
- Berdichevsky, V., "Mechanical Equations of Particulate Fluids," (survey in "Problems of Averaging and Continuum Models Development"), MGU Publishing, Moscow, pp. 10-35, 1980
- Berdichevsky, V., "Variational Principles in the Problem of Averaging Random Structures," Soviet Physics - Doklady, Vol. 26, No. 11, pp. 1053-1055, 1981.
- Berdichevsky, V. and Sutyrin, V., "The Dynamics of Periodic Structures," Soviet Physics - Doklady, Vol. 28, No. 3, pp. 239-241, 1983.
- Berdichevsky, V. and Sutyrin, V., "Problem of an Equivalent Rod in Nonlinear Theory of Springs," J. Appl. Math. Mech. (PMM), Vol. 47, No. 2, pp. 197-205, 1983.
- Berdichevsky, V., "Heat Conduction of Chess Structures," Mechanical Bulletin Moscow University, pp. 15-25, 1985.
- Berdichevsky, V., "The Problem of Averaging Random Structures in Terms of Distribution Functions," J. Appl. Math. Mech. (PMM), Vol. 51, No. 6, pp. 704-710, 1987.
- Berdichevsky, V., "Distribution of functionals of solutions in some stochastic variational problems," Yudovich jubilee volume, Izv.Vuz.,56-59,2004

**STATISTICAL MECHANICS**

- Berdichevsky, V., Thermodynamics of Chaos and Order, Addison Wesley Longman, 267 pp., 1997.
- Berdichevsky, V., Distribution of minimum values of weekly stochastic functionals, in Homogenization, V.Berdichevsky, V.Jikov, G.Papanicolaou, eds.,World Scientific, pp.141-186,1999.
- Berdichevsky, V., "A Connection Between Thermodynamic Entropy and Probability," J. Appl. Math. Mech. (PMM), Vol. 52, No. 6, pp. 947-957, 1988.
- Berdichevsky, V., "Law of Evolution to Equilibrium in Nonlinear Thermodynamics," Int. J. of Eng. Sci., Vol. 28, No. 7, pp. 697-704, 1990.
- Berdichevsky, V., and von Alberti, M., "Statistical Mechanics of Hénon-Heiles Oscillators," Physical Review A, Vol. 44, p. 858-865, 1991.
- Berdichevsky, V., "Generalized Equipartition Law," Int. J. of Engng. Sci., Vol. 31, No. 4, pp. 673-677, 1993.
- Berdichevsky, V., "Reciprocal Relations in Nonlinear Vibrations," Int. J. of Engng. Sci., Vol. 31, No. 8, pp. 1215-1218, 1993.
- Berdichevsky, V., Ozbek, A., Kim W.W., Thermodynamics of Duffing's Oscillator, J.of Applied Mechanics , Vol.61, pp.670-675, 1994.
- Berdichevsky, V., Kim W.W., Ozbek,A., Dynamical Potential for Nonlinear Vibrations of Cantilevered Beam, J.Sound and Vibrations ,vol.179,pp.151-164,1995.
- Berdichevsky, V., Le, K.C., A theory of charge nucleation in two dimensions,Phys.Rev.E,026129(August),17p.,2002
- Berdichevsky, V., Structure of equations of macrophysics, Physical Review E,vol.68,066126,26p.,2003.
- Berdichevsky, V., Distribution of functionals of solutions in some stochastic variational problems, Yudovich's anniversary volume, Mathematics and mechanics of continuum media, Izv.Vuz., 56-59, 2004.

**TURBULENCE**

- Berdichevsky, V., "On Turbulence Theory," in the Anniversary volume for the 80th Birthday of Academician L. Sedov, Moscow University Publishing, Moscow, 1987.
- Berdichevsky, V., Statistical Mechanics of Point Vortices, Physical Review E ,vol.51,No.5,4432-4452,1995.
- Berdichevsky, V., Fridlyand, A., Sutyrin, V., Prediction of Turbulent Velocity Profile in Couette and Poiseuille Flows from First Principles, Physical Review Letters, vol.76,#21, 3967-3970, 1996.
- Berdichevsky, V., Statistical Mechanics of Vortex Lines, Physical Review E, v.57, No.3, 2885-2905,1998.
- Le K.C.,Berdichevsky V.L.,Energy distribution in a neutral gas of point vortices, J.Statistical Physics,v.104,883-890,2001.
- Berdichevsky V.L., On statistical mechanics of vortex lines,Int.J.Engng. Sci, v.40,123- 129,2002.
- Averaged equations of ideal fluid turbulence, Continuum Mechanics and Thermodynamics, v.19, 133-175, 2007.
- On statistical mechanics of vortex lines . IUTAM Symposium on Hamiltonian dynamics, vortex structures and turbulence. Springer-Verlag, 205-210, 2008.

**VARIATIONAL PRINCIPLES**

- Berdichevsky, V., Variational Principles of Continuum Mechanics, Publishing House "Nauka" of the Academy of Science of the USSR, 446 pp., 1983.
- Berdichevsky, V., "Construction of Models of Continuous Media by Means of the Variational Principle," J. Appl. Math. Mech. (PMM), Vol. 30, No. 3, pp. 607-629, 1966
- Berdichevsky, V., "Use of Variational Methods in Constructing Models of Continuous Media with Irreversible Processes in the Special Theory of Relativity," J. Appl. Math. Mech. (PMM), Vol. 30, No.6, pp. 1282-1287, 1966.
- Berdichevsky, V., "Variational Equation of Continuum Mechanics," in: Problems of Solid Mechanics, Anniversary volume for the 60th Birthday of Academician V. Novozhilov, Sudostroyeniye, Leningrad, 1970.
- Berdichevsky, V., "A Variational Principle," Soviet Physics - Doklady, Vol. 19, No. 4, pp. 188-190, 1974.
- Sedov, L. and Berdichevsky, V., "Extracting of Real Processes from the Optional Ones," in : Science and Mankind, Znanie, Moscow, 1976.
- Berdichevsky, V., "Variational-Asymptotic Method," in: Some Problems of Continuum Mechanics, Anniversary volume for the 70th Birthday of Academician L. Sedov, Moscow University Publishing, Moscow, pp. 271-289, 1978.
- Berdichevsky, V. and Misyura, V., "On a Dual Variational Principle in Geometrically Nonlinear Elasticity Theory," J. Appl. Math. Mech., Vol. 43, No. 2, pp. 343-351, 1979.
- Berdichevsky, V., "Variational Principles for Distribution Function," in: Problems of Nonlinear Continuum Mechanics, Tallin, 1985.
- Berdichevsky, V., "On a Variational Principle of Statistical Mechanics," Mechanical Bulletin of Moscow University, pp. 8-14, 1987.
- Variational principle in dynamics of a vortex filament. Physical Review E 78, 036304 2008.
- On variational features of vortex flows. Continuum Mechanics and Thermodynamics (2008) 20: 219.229 DOI 10.1007/s00161-008-0079-8

**SHELL THEORY**

- Berdichevsky, V., "Variational Methods on Constructing Models of Shells," J. Appl. Math. Mech., Vol. 36, No. 5, pp. 743-757, 1972.
- Berdichevsky, V., "Equations Describing Transverse Vibration of Thin Elastic Plates," Mechanics of Solids, No. 6, pp. 138-141, 1972.
- Berdichevsky, V., "Dynamic Theory of Thin Elastic Plates," Mech. of Solids, No. 6, pp. 86-96, 1973
- Berdichevsky, V., "An Energy Inequality in the Theory of Plate Bending," J. Appl. Math. Mech., Vol. 37, No. 5, pp. 891-895, 1973.
- Berdichevsky, V., "Dynamic Equations in the Theory of Anisotropic Plates," Soviet Physics -Doklady, Vol. 20, No. 9, pp. 643-645, 1975.
- Berdichevsky, V., "Stress Functions and Some A Priori Estimates in Plate Bending Theory," J. Appl. Math. Mech., Vol. 40, No. 3, pp. 481-488, 1976.
- Berdichevsky, V. and Koc, L., "On Equation Describing the Transverse Vibrations of Anisotropic Elastic Plates," in Selected Problems of Applied Mechanics, Anniversary volume for the 60th Birthday of Academician V. Chalomey, Academy of Science of USSR, Moscow, 1974.
- Berdichevsky, V., "Some Forms of the Equations of Shell Theory," Soviet Physics - Doklady, Vol. 22, No. 4, pp. 231-233, April 1977.
- Berdichevsky, V., "High Frequency Long Wave Vibrations of Plates," Soviet Physics - Doklady, Vol. 22, No. 10, pp. 604-606, October 1977.
- Berdichevsky, V., "On the Validation of the Membrane Theory of Shells," Mechanics of Solids, No. 6, pp. 148-152, 1977.
- Berdichevsky, V., "Variational-Asymptotic Method of Constructing a Theory of Shells," J. Appl. Math. Mech., Vol. 43, No. 4, pp. 711-736, 1979.
- Berdichevsky, V. and Le, Khanh Chau, "High Frequency Long Waves Shell Vibration," J. Appl. Math. Mech. (PMM) , Vol. 44, No. 4, pp. 520-525, 1980.
- Berdichevsky, V. and Le, Khanh Chau, "High Frequency Vibrations of Shells," Soviet Physics -Doklady, Vol. 27, No. 11, pp. 988-990, November 1982.
- Berdichevsky, V., "Tensor Form of Novozhilov-Babaluch Equations," Mechanics of Solids, No. 4, pp. 167-170, 1982.
- Berdichevsky, V., and Misyura, V., "Effect of Accuracy Loss in Classical Shell Theory," Journal of Applied Mechanics, Vol. 59, No. 2, Pt. 2, pp. 5217-5223, 1992.
- An asymptotic theory of sandwich plates. International Journal of Engineering Science. doi:10.1016/j.ijengsci.2009.09.001, 2009
- Nonlinear theory of hard-skin plates and shells. International Journal of Engineering Science. doi:10.1016/j.ijengsci.2009.09.003

**BEAM THEORY**

- Berdichevsky, V. and Kvashnina, S., "On Equations Describing the Transverse Vibrations of Elastic Bars," J. Appl. Math. and Mech., Vol. 40, No. 1, pp. 104-118, 1976.
- Berdichevsky, V., "On Equations of the Theory of Anisotropic Inhomogeneous Rods," Soviet Physics - Doklady, Vol. 21, No. 6, pp. 286-288, May 1976.
- Berdichevsky, V. and Staroselsky, L., "On the Theory of Naturally Twisted Curvilinear Rods," Mechanics of Solids, No. 6, pp. 87-95, 1979.
- Berdichevsky, V., "On the Energy of an Elastic Beam," J. Appl. Math. Mech. (PMM), Vol. 45, No. 4, pp. 518-529, 1981.
- Berdichevsky, V., "Dispersion of Nonlinear Waves in a Rod," J. Appl. Math. Mech., Vol. 46, No. 2, pp. 244-249, 1982.
- Berdichevsky, V. and Staroselsky, L., "On the Theory of Curvilinear Timoshenko-Type Rods," J. Appl. Math. Mech., Vol. 47, No. 6, pp. 809-816, 1983.
- Berdichevsky, V. and Staroselsky, L., "Bending, Extension and Torsion of Naturally Twisted Rods," J. Appl. Math. Mech., Vol. 49, No. 6, pp. 746-755, 1985.
- Berdichevsky, V., Armanios, E., and Badir, A., "Theory of Thin-Walled Anisotropic Beams," Journal of Composite Materials, Vol. 2, No. 5-7, pp. 411-432, 1992.
- Volovoi, V., Hodges, D., Berdichevsky, V., Sutyrin, V.,Dynamic dispersion curves for nonhomogeneous anisotropic beams with cross sections of arbitrary geometry, J.Sound and Vibrations, v.115,N.5, 1101-1120,1998.
- Volovoi, V., Hodges, D., Berdichevsky, V., Sutyrin, V., Asymptotic theory for static behavior of elastic anisotropic I-beams, Int.J.Solids and Structures, v.36, 1017-1043,1999.

**SAINT-VENANT'S PRINCIPLE**

- Berdichevsky, V., "To the Proof of the Saint-Venant Principle for Bodies with Arbitrary Shape," J. Appl. Math. Mech., Vol. 38, No. 5, pp. 799-811, 1974.
- Berdichevsky, V., "Energy Methods in Some Problems of Decay of Solutions," J. Appl. Math. Mech., Vol. 42, No. 1, pp. 140-156, 1978.
- Berdichevsky, V., "Decay Rate of Stresses in Cylindrical Elastic Bodies," Soviet Physics -Doklady, Vol. 21, No. 9, pp. 534-536, September 1976.
- Berdichevsky, V.,Foster, D.J., On Saint-Venant's Principle in the Dynamics of Elastic Beams, Int. J. Solids and Structures, v.40, 3293-3310, 2003.
- Foster, D.J., Berdichevsky, V., On Saint-Venant's principle in the two-dimensional flexural vibrations of elastic beams, Int. J. Solids and Structures, v.41, 2551-2562,2004

**MISCELLANEOUS**

- Berdichevsky, V., "On the Force Acting on a Body in Viscous Fluids," J. Appl. Math. Mech.(PMM), Vol. 45, No. 5, pp. 628-631, 1981.
- Berdichevsky, V., "On Nonlinear Tensor Functions in Relativity Theory "J. Appl. Math. Mech. (PMM), Vol. 31, No. 1, pp. 203-206, 1967.
- Berdichevsky, V. and Berdichevsky, A., "Proof of the Polia Hypothesis," Soviet Physics - Doklady, Vol. 20, No. 9, pp. 614-615, 1975.
- Berdichevsky, V., "Form of Minimum-Drag Body in Hypersonic Gas Flow," Moscow University Mechanics Bulletin, Vol. 30, No. 3-4, pp. 13-18, 1975.
- Berdichevsky, V., "Melt Nuclei in a Solid," Transactions of Academy of Science of the USSR, pp. 1-4, 1984.
- Berdichevsky, V., "Determination of the Width of a Boundary Layer from the Condition of Minimum Energy," Mechanics of Solids, No. 2, pp. 173-176, 1983.
- Feynman, J.,Ruzmaikin,A., Berdichevsky, V.,The JPL proton fluence model: an update, J. of Atmospheric and Solar-Terrestrial Physics,p.1-8,2000