FUNDAMENTAL FACTS FROM THE LINEAR THEORY OF ELASTICITY
Note 3.15. Two basic variational methods are used here – the Lagrange’s principle based on variations of displacements and the Castigliano’s principle based on
variations of stresses.
The mentioned approximate numerical methods are actively used and modified.
There are many commercial computer packages for solving the big classes of problems.
Comments
Comment 3.1. The proofs of theorems and rigorous reasonings have an important goal of removing the doubts. Only a professional mathematician can enjoy the formal justification of each step of a long line of reasoning.
As for the removal of doubts, there is a story about D’Alembert:
Unsuccessfully explaining the proof of some theorem to one noble pupil, he said: Sir!
Honestly, this theorem is right! The reaction of the noble pupil was instant: Oh, sir! That’s enough. You’re a nobleman and I’m a nobleman. And your honest word is the best of proof.
Comment 3.2. In this chapter, many facts from the linear theory of elasticity are formulated in some abstract form. This contradicts in some cases to the understanding of the mechanics of materials as a science having also the function of being useful for engineers. Sometimes they joke that the presentation of the solution in the form 7arctan2 2 7ln
11 5 3 6 is not acceptable for engineers. It should have to engineer the form 0.338.
FOUNDATIONS OF MECHANICS OF MATERIALS PART 1
SOME ADDITIONAL FUNDAMENTAL FACTS FROM THE LINEAR THEORY OF ELASTICITY
Further reading
3.1. Achenbach, JD 1973, Wave Propagation in Elastic Solids. North-Holland, Amsterdam.
3.2. Ang, W-T 2007, A Beginner’s Course in Boundary Element Methods. Universal Publishers, Boca Raton, USA.
3.3. Ascione, L & Grimaldi, A 1993, Elementi di meccanica del continuo (Elements of Continuum Mechanics). Massimo, Napoli. (In Italian)
3.4. Atkin, RJ & Fox, N 1980, An Introduction to the Theory of Elasticity, Longman, London.
3.5. Banerjee, PK 1994, The Boundary Element Methods in Engineering. 2nd ed., McGraw- Hill, London.
3.6. Bedford, A & Drumheller, DS 1994, Introduction to Elastic Wave Propagation. John Wiley, Chichester.
3.7. Chaskalovic, J 2008, Finite Elements Methods for Engineering Sciences, Springer Verlag, Berlin.
3.8. Chen, PJ 1972, Wave Motion in Solids. Flügge‘s Handbuch der Physik, Band VIa/3.
Springer Verlag, Berlin.
3.9. Dagdale, DS & Ruiz, C 1971, Elasticity for Engineers. McGraw Hill, London.
310. Den Hartog JP 2007, Mechanical Vibrations, 12th ed. Dover Civil and Mechanical Engineering, Mineola.
3.11. Eschenauer, H & Schnell, W 1981, Elasticitätstheorie I, Bibl. Inst., Mannheim. (In German)
3.12. Eslami, MR, Hetnarski, RB, Ignaczak, J, Noda, N, Sumi, N & Tanigawa, Y 2013, Theory of Elasticity and Thermal Stresses. Explanations, Problems and Solutions. Series Solid Mechanics and Its Applications, Springer Verlag Netherlands, Amsterdam.
3.13. Fedorov, FI 1968, Theory of Elastic Waves in Crystals. Plenum Press, New York.
3.14. Fraeijs de Veubeke, BM 1979, A Course of Elasticity. Springer, New York.
3.15. Graff, KF 1991, Wave Motion in Elastic Solids. Dover, London.
3.16. Harris, JG 2001, Linear Elastic Waves. Cambridge Texts in Applied Mathematics.
Cambridge University Press, Cambridge.
3.17. Hudson, JA 1980, The Excitation and Propagation of Elastic Waves. Cambridge University Press, Cambridge.
3.18. Hahn, HG 1985, Elastizitätstheorie. B.G.Teubner, Stuttgart. (In German) 3.19. Inman, DJ 2007, Engineering Vibration, 3rd ed. Prentice Hall, New York.
3.20. Katsikadelis, JT 2002, Boundary Elements Theory and Applications. Elsevier, Amsterdam.
3.20. LeVeque, RJ 2007, Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM, New York.
3.21. Love, AEH 1944, The Mathematical Theory of Elasticity. 4th ed. Dover Publications, New York.
3.22. Lur’e AI 1999, Theory of Elasticity. Series “Foundations of Engineering Mechanics”.
Springer, Berlin.
3.23. Magnus K 1976, Schwingungen. Eine Einfuhrung in die theoretische Behandlung von Schwingugsprobleme. Teubner, Stuttgart. (In German)
3.24. Müller, W 1959, Theorie der elastischen Verformung. Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig. (In German)
3.25. Nowacki, W 1970, Theory of Elasticity. PWN, Warszawa. (In Polish, In Russian) 3.26. Olver, P 2013, Introduction to Partial Differential Equations. Chapter 5: Finite
differences. Springer, Berlin. 3.27. Reddy, JN 2006, An Introduction to the Finite Element Method. 3rd ed., McGraw-Hill, New York.
3.28. Reynolds, DD 2016, Engineering Principles of Mechanical Vibration, 4th ed. Trafford On Demand Publishing, Bloomington, USA.
3.29. Royer, D & Dieulesaint, E 2000, Elastic Waves in Solids (I,II). Advanced Texts in Physics. Springer, Berlin.
3.30. Rushchitsky, JJ 2011, Theory of Waves in Materials. Ventus Publishing ApS, Copenhagen.
3.31. Slaughter, WS 2001, Linearized Theory of Elasticity. Birkhäuser, Zurich.
3.32. Sneddon, IN & Berry, DS 1958, The Classical Theory of Elasticity, vol.VI, Flügge Encyclopedia of Physics. Springer Verlag, Berlin.
3.33. Sokolnikoff, IS 1956, Mathematical Theory of Elasticity. McGraw Hill Book Co, New York.
3.34. Starovoitov, E & Naghiyev, FBO 2012, Foundations of the Theory of Elasticity, Plasticity, and Viscoelasticity, Apple Academic Press, Palo Alto.
3.35. Strikwerda, J 2004, Finite Difference Schemes and Partial Differential Equations. 2nd ed. SIAM, New York.
3.36. Timoshenko, SP & Goodyear, JN 1970, Theory of Elasticity, 3rd ed. McGraw Hill, Tokyo.
3,37. Tongue, BH 2001, Principles of Vibration, Oxford University Press, Oxford.
3.38. Wrobel, LC & Aliabadi, MH 2002, The Boundary Element Method. In 2 vols. John Wiley & Sons, New York.
3.39. Zienkiewicz, OC, Taylor, RL & Zhu, JZ 2005, The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, New York.
Questions
3.1. Estimate the role of inverse problems in the linear theory of elasticity. Compare with the importance of the inverse problems in geophysics.
3.2. Read something about the existence theorem. Find two Korn’s inequalities and Fichera’s proof of this theorem.
3.3. Show the link among the different general representation of solutions.
3.4. Find and describe the solved simple problem within the framework of the anti-plane
FOUNDATIONS OF MECHANICS OF MATERIALS PART 1
SOME ADDITIONAL FUNDAMENTAL FACTS FROM THE LINEAR THEORY OF ELASTICITY
3.5. Repeat the step-by-step procedure of proving that the generalized plane stress state is identical with the plane displacement state.
3.6. Comment an importance of the Airy’s function.
3.7. Find a few handbooks on the stress concentration around holes, necks, cuts and formulate the most important applications of results on the stress concentration.
3.8. Describe shortly the fundamentality of universal deformations in the theory of elasticity.
3.9. Choose the most convenient for you book devoted to rods and beams.
3.10. Choose the most convenient for you book devoted to plates and shells.
3.11. Which modern book on vibrations of the elastic body you prefer?
3.12. Look for the modern books on plates and shells and choose one book on plates and one book on shells, which you could recommend your colleagues.
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