Abstract
A new concept in mechanics is proposed based on the notions of space, time, and energy. The energy is represented by the sum of thirteen terms expressed as products of invariants of the Lagrangian equations of motion and scalar factors corresponding to the physical properties of materials. Differential equations of motion and equilibrium are derived from the frame indifference condition for energy, in which the Eulerian variables are the sought functions and the Lagrangian variables are the arguments. Assumptions are made under which these equations can be transformed into the Poisson and Laplace equations. The law of conservation of energy is used to obtain dependences between the Lagrangian stresses and strains in the reversible deformation region; they are compared with the expressions used in the theory of elasticity. It is discussed whether the point of reference for the mean stresses may be chosen with regard to the volume energy density of particles in their initial state, as well as whether elastic deformation can be described using a single constant. As an example, the energy model and the equations of motion in the Lagrangian form are applied to describe the transition from reversible to irreversible deformation. The developed model differs from the classical one by using two independent infinitesimal operators for the time and space. It is shown that Newton’s law of inertia can be regarded as a variant for the determination of the generalized forces characterizing the kinetic energy change in a body with increments in the distance between the origin of the observer coordinate system and the center of mass of the body. The application of the Lagrangian variables and the superposition principle for describing any spatial motions (including for absolutely rigid bodies), which can be used in the dynamic analysis of linkages and other mechanisms, is validated. The multiple choice of generalized forces for absolutely rigid bodies, including when passive forces arise, is considered. The method of dynamic analysis of mechanisms developed on the basis of the energy model allows the law of conservation of energy to be satisfied for any part of the studied system in an arbitrary time interval.
Similar content being viewed by others
References
Sedov, L.I., A Course in Continuum Mechanics, Groningen, Wolters-Noordhoff, 1971.
Kolmogorov, V.L., Stresses, Deformations, Fracture, Moscow: Metallurgiya, 1970.
Alyushin, Yu.A., Energy Foundations of Mechanics: A Handbookfor Students, Moscow: Mashinostroenie, 1999.
Alyushin, Yu.A., Defining Relations of Reversible and Irreversible Deformation in the Lagrangian Description, J. Machinery Manufact. Reliabil., 2007, no. 5, pp. 434–442.
Hill, R., The Mathematical Theory of Plasticity, Oxford: Clarendon Press, 1950.
Alyushin, Yu.A., Energy Scale of the Average Stresses and Physical Properties of Metals in the Region of Reversible and Irreversible Deformations, J. Machinery Manufact. Reliabil., 2010, no. 3, pp. 282–289.
Korn, G.A. and Korn, T.M., Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, Mineola, New York: Dover Publications, 2000.
Ishlinsky, A.Yu., Mechanics: Ideas, Problems, Applications, Moscow: Nauka, 1985.
Courant, R. and Hilbert, D., Methods of Mathematical Physics: Partial Differential Equations, New York: Interscience Publishers, 1962.
Alyushin, Yu.A., Solid Mechanics in Lagrangian Coordinates: A Handbook for Students, Moscow: Mashinostroenie, 2012.
Kaye, G.W.C. and Laby, T.H., Tables of Physical and Chemical Constants and Some Mathematical Functions, Longman, London, 1973.
Landau, L.D. and Lifshitz, E.M., Course of Theoretical Physics: Statistical Physics, V. 5, Pergamon Press, Oxford, 1980.
Alyushin, Yu.A., The Principle of Superposition of Motions in the Space of Lagrangian Coordinates, Probl. Mashinostroen. Nadezhn. Mashin, 2001, no. 3, pp. 13–19.
Alyushin, Yu.A., Force Calculation for an Articulated Linkage Based on Energy Flow Analysis, Probl. Mashinostroen. Nadezhn. Mashin, 2003, no. 2, pp. 125–133.
Ishlinsky, A.Yu., On Absolute Forces and Inertia Forces in Classical Mechanics.http://termech.mpei.ac.ru
Alyushin, Yu.A., Energy Nature of Centrifugal and Newtonian Forces, Int. J. Mech. Eng. Automat., 2016, vol. 3, no. 3, pp. 121–127.
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2018, published in Fizicheskaya Mezomekhanika, 2018, Vol. 21, No. 3, pp. 59–69.
Rights and permissions
About this article
Cite this article
Alyushin, Y.A. A New Concept in Mechanics Based on the Notions of Space, Time, and Energy. Phys Mesomech 22, 536–546 (2019). https://doi.org/10.1134/S1029959919060109
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1029959919060109