THEORY OF ELASTICITY
Note 2.7. The ergodicity theorem is physically meaningless if the characteristic time of the system is commensurate (longer) with the time it takes to determine
the average values used in the theorem.
It is believed that the average over the ensemble of functions characterizing a discrete system are indeed observable quantities. This allows us to move on to the macroscopic characteristics of the system and thereby use the concept of the continuum.
In other words, if a macroscopically very small volume of a substance with a large number of microscopically small particles (molecules) is considered and a system with a macroscopically small time exceeding the characteristic time of the system is observed, then the system is equilibrium one and the ergodicity theorem holds for it.
It is considered that statistical interpretations help in understanding the physical meaning in the formal transition from a discrete model of matter to a continuum.
Comment 2.2. Let us start with the stated in this chapter sentence that the modern interpretation of research on solid mechanics relates them to three modern sections of science - applied mathematics, mechanical and other engineering, material science. In fact, the modern mechanics, as a science and a profession, have disappeared into the three modern
sections of science mentioned above. The vast majority of universities in the world do not train specialists in mechanics. At the same time, about 60 developed countries of the world have the national organizations of mechanicians and are united in the International Union of Theoretical and Applied Mechanics (IUTAM) and European mechanicians are united in the European Mecha-nics Society (EuroMech). Each of three sciences - applied mathematics, mechanical and other enginee-ring, material science - treat mechanics in its way. For example, at one of the International Congress of Industrial and Applied Mathematics (ICIAM), the plenary lecturer formulated the essence of applied mathematics in this way: There is no special science applied mathematics but applied mathematicians nonetheless exist. These are specialists who use the achievements of mathematics for non-mathematical purposes, allowing the use of non-mathematical means to justify their actions.
From this point of view, the assignment of mechanics to applied mathematics, perhaps, reduces the prestige of mechanics as fundamental science.
Further reading
2.1. Ascione, L & Grimaldi, A 1993, Elementi di meccanica del continuo (Elements of continuum mechanics). Mas simo, Napoli. (In Italian)
2.2. Ashby, MF 2005, Materials Selection in Mechanical Design, 3rd ed., Elsevier, Amsterdam- Tokyo.
2.3. Atkin, RJ & Fox, N 1980, An Introduction to the Theory of Elasticity, Longman, London.
2.4. Bell, JF 1973, Experimental foundations of solid mechanics. Flugge’s Handbuch der Physik, Band VIa / 1, pringer Verlag, Berlin.
2.5. Dagdale, DS & Ruiz, C 1971, Elasticity for Engineers. McGraw Hill, London.
2.6. Ericksen, JL 1998, Introduction to the Thermodynamics of Solids, Applied Mathematical Sciences, vol. 131, Springer, Berlin
2.7. Eringen, AC 1967, Mechanics of Continua. John Wiley, New York.
2.8. Eschenauer, H & Schnell, W 1981, Elasticitätstheorie I, Bibl. Inst., Mannheim. (In German)
2.9. Eslami, MR, Hetnarski, RB, Ignaczak, J, Noda, N, Sumi, N & Tanigawa, Y 2013, Theory of Elasticity and Thermal Stresses. Explanations, Problems and Solutions. Springer Series “Solid Mechanics and Its Applications,Springer Verlag Netherlands, Amsterdam.
2.10. Fraeijs de Veubeke, BM 1979, A Course of Elasticity. Springer, New York.
2.11. Fu, YB 2001, Nonlinear Elasticity: Theory and Applications. London Mathematical Society Lecture Note Series,Cambridge University Press, Cambridge.
2.12. Fung, YC 1965, Foundations of Solid Mechanics. Prentice Hall, Englewood Cliffs.
2.13. Germain, P 1973, Cours de mécanique des milieux continus. Tome 1. Théorie Générale.
Masson et Cie Editeurs, Paris. (In French)
FOUNDATIONS OF MECHANICS OF MATERIALS PART 1
BASIC INFORMATION ON MECHANICS OF MATERIALS.
THEORY OF ELASTICITY. SHORT DESCRIPTION OF LINEAR THEORY OF ELASTICITY
2.15. Hahn, HG 1985, Elastizitätstheorie. B.G.Teubner, Stuttgart. (In German)
2.16. Holzapfel, GA 2000, Nonlinear Solid Mechanics: A Continuum Approach for Engineering.
Birkhauser, Zurich.
2.17. Iliushin, AA 1990, Mechanics of Continuum, Moscow University Publishing House, Moscow. (In Russian)
2.18. Johns, DJ 1965, Thermal Stress Analysis, Pergamon Press, Oxford.
2.19. Kobayashi, AS (ed) 1987, Handbook on experimental mechanics. Prentice-Hall, Englewood Cliffs.
2.20. Korotkina, MR 1988, Elektromagnetoelasticity, Moscow University Press, Moscow. (In Russian)
2.21. Love, AEH 1944, The Mathematical Theory of Elasticity. 4th ed. Dover Publications, New York.
2.22. Lur’e AI 1999, Theory of Elasticity. Springer Series in Foundations of Engineering Mechanics. Springer, Berlin.
2.23. Lur’e, AI 1990, Nonlinear Theory of Elasticity. North-Holland Series in Applied Mathematics and Mechanics, North-Holland, Amsterdam.
2.24. Maugin, GA 1988, Continuum Mechanics of Electromagnetic Solids. North Holland, Amsterdam.
2.25. Müller, W 1959, Theorie der elastischen Verformung. Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig. (In German)
2.26. Nowacki, W 1962, Thermoelasticity. Pergamon Press, Oxford.
2.27. Nowacki, W 1970, Theory of Elasticity. PWN, Warszawa. (In Polish, In Russian) 2.28. Podstrigach, JS & Povstenko, YZ 1985, Introduction into mechanics of surface phenomena
in solids, Naukova Dumka, Kiev. (In Russian)
2.29. Prager, W 1961, Introduction to Mechanics of Continua. Ginn, Boston.
2.30. Ratner, LW 2003, Non-Linear Theory of Elasticity and Optimal Design. Elsevier, London.
2.31. Rushchitsky, JJ & Tsurpal, SI 1998, Waves in Materials with the Microstructure. S.P.
Timoshenko Institute of Mechanics, Kiev. (In Ukrainian)
2.32. Savin, GN & Rushchitsky, JJ 1976, Elements of Mechanics of Hereditary Media. Vyshcha Shkola, Kyiv. (In Ukrainian)
2.33. Sedov, LI 1970, Mechanics of Continuum, in 2 vols. Nauka, Moscow. (In Russian) 2.34. Slaughter, WS 2001, Linearized Theory of Elasticity. Birkhauser, Zurich.
2.35. Sneddon, IN & Berry, DS 1958, The Classical Theory of Elasticity, vol.VI, Flügge Encyclopedia of Physics. Springer Verlag, Berlin.
2.36. Sokolnikoff, IS 1956, Mathematical Theory of Elasticity. McGraw Hill Book Co, New York.
2.37. Sommerfeld, A 1964, Thermodynamics and Statistical Mechanics. Academic Press, New York.
2.38. Spencer, AJM 1980, Continuum Mechanics. Longman, London.
2.39. Starovoitov, E & Naghiyev, FBO 2012, Foundations of theTheory of Elasticity, Plasticity, and Viscoelasticity, Apple Academic Press, Palo Alto.
2.40. Storakers, B & Larsson, P-L 1998, Introduktion till finit elasticitetteori. Hallfasthetslara, KTH. (In Swedish)
2.41. Taber, LA 2004, Nonlinear Theory of Elasticity: Applications in Biomechanics.
Birkhauser, Zurich.
2.42. Timoshenko, SP & Goodyear, JN 1970, Theory of Elasticity, 3rd ed. McGraw Hill, Tokyo.
2.43. Truesdell, C 1969, Rational Thermodynamics. McGraw-Hill Book Company, New York.
2.44. Truesdell, C 1972, A First Course in Rational Continuum Mechanics. The John Hopkins University, Baltimore.
Questions
2.1. Which more complicated combinations of mechanical properties of materials (for example, the property of gyroelasticity) exist, occurring in the real practice and reflecting in the me chanical theories and do not mention in the chapter? Indicate the degree of development of such theories.
2.2. Is the property of viscoelasticity characteristic for materials only? By other words, it is possible to speak about the viscoelastic materials and the viscoelastic fluids?
2.3. Formulate similarity and distinction between rheology and viscoelasticity.
2.4. Presence of which properties will need the attraction of thermodynamical considerations when the mechanical model being created?
2.5. Which property of material causes energy dissipation when the material being deformed?
2.6. Find the not mentioned here classifications of mechanics and compare them with the basic ones.
2.7. Which text-book on the theory of elasticity you prefer and which book you would add to the proposed list above? Formulate the advantages of preferred book.
FOUNDATIONS OF MECHANICS OF MATERIALS PART 1
SOME ADDITIONAL FUNDAMENTAL FACTS FROM THE LINEAR THEORY OF ELASTICITY