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Continuum Mechanics – Material Independent and Dependent Equations

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Lectures Notes on Advanced Structured Materials 2

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 203))

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Abstract

Continuum mechanics in its rational formulation is a central element within mechanics. The basic concepts can be easily transferred to many special cases, with kinematics, stress models, balance equations, and constitutive laws being the most important elements of the theories. It also applies here that statics is the special case in Newton’s sense. Selected historical stages are briefly described below. The three-dimensional continuum theory is then briefly discussed. Finally, a note on the one- and two-dimensional special cases is given. The chapter is written from the point of view of a solid-state mechanics. However, numerous equations can easily be transferred to fluids.

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Notes

  1. 1.

    Isaac Todhunter (\(^*\)23 November 1820, Rye, Sussex, England, \(\dag \)1 March 1884, Cambridge, Cambridgeshire, England) was an English mathematician and well known for books on the history of sciences. He was elected as a Fellow of the Royal Society.

  2. 2.

    Karl Pearson (born as Carl Pearson, \(^*\)27 March 1857, Islington, London, England, \(\dag \)27 April 1936, Coldharbour, Surrey, England) was an English mathematician and biostatistician. He established the mathematical statistics and was elected as a Fellow of the Royal Society and a Fellow of the Royal Society of Edinburgh. He was a staunch supporter of eugenics.

  3. 3.

    About \(^*\)287–\(\dag \)212 BC, Syracus, Sicily.

  4. 4.

    \(^*\)18 July 1635, Freshwater, Isle of Wight, England, \(\dag \)3 March 1703, London, England, English polymath being a scientist, natural philosopher, and architect. He was one of the first scientists discovering microorganisms with a help of a compound microscope built himself. He was a member of the Royal Society and since 1662 its curator of experiments.

  5. 5.

    English translation “As the extension, so the force”.

  6. 6.

    The concept was developed in 1727 by Leonhard Euler and the first experiments that used the concept of Young’s modulus in its current form were performed by the Italian Giordano Riccati (\(^*\)25 February 1709, Castelfranco, Veneto, \(\dag \)20 July 1790, Treviso, Veneto, contributions to music, architecture, mechanics, and mathematics.) in 1782 – predating Young’s work by 25 years.

  7. 7.

    \(^*\)4 January 1643, Woolsthorpe-by-Colsterworth, Lincolnshire, England, \(\dag \)31 March 1727, Kensington, Middlesex, Great Britain, English mathematician, physicist, astronomer, alchemist, theologian, 12\(\textrm{th}\) President of the Royal Society.

  8. 8.

    \(^*\)15 April 1707, Basel, Swiss Confederacy, \(\dag \)18 September 1783, Saint Petersburg, Russian Empire, mathematician, physicist, astronomer, geographer, logician, and engineer working in Switzerland, Prussia, and in the Russian Empire.

  9. 9.

    Leonardo di ser Piero da Vinci (\(^*\)15 April 1452, possibly Anchiano, Vinci, Republic of Florence, \(\dag \)2 May 1519 (aged 67) Clos Lucé, Amboise, Kingdom of France), Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect.

  10. 10.

    \(^*\)6 January 1655, Basel, Switzerland, \(\dag \)16 August 1705, Basel, Switzerland, mathematician with contributions to mechanics.

  11. 11.

    \(^*\)21 August 1789, Paris, France, \(\dag \)3 May 1857, Sceaux, France, French mathematician, engineer, and physicist with pioneering contributions to continuum mechanics.

  12. 12.

    \(^*\)13 June 1773, Milverton, Somerset, England, \(\dag \)10 May 1829, London, England, British polymath with contributions to the fields of light, solid mechanics, physiology, language, musical harmony, and Egyptology.

  13. 13.

    \(^*\)21 June 1781, Pithiviers, Kingdom of France (present-day Loiret), \(\dag \)25 April 1840, Sceaux, Hauts-de-Seine, Kingdom of France, French mathematician and physicist.

  14. 14.

    Gabriel Lamé (\(^*\)22 July 1795, Tours, \(\dag \)May 1870, Paris), French mathematician.

  15. 15.

    \(^*\)11 February 1839, New Haven, Connecticut, U.S., \(\dag \)28 April 1903, New Haven, Connecticut, U.S., American scientist who made significant theoretical contributions to physics, chemistry, and mathematics.

  16. 16.

    \(^*\)26 November 1852, Douai, France, \(\dag \)22 March 1914, France, French engineer and mathematician.

  17. 17.

    \(^*\)4 March 1866, Amiens, France, \(\dag \)31 May 1931, Toulouse, France, French mathematician and astronomer.

  18. 18.

    Jerald LaVerne Ericksen (\(^*\)20 December 1924, Portland, Oregon, USA, \(\dag \)11 June 2021, Minisota, USA), American mathematician specializing in continuum mechanics.

  19. 19.

    Clifford Ambrose Truesdell III (\(^*\)18 February 1919, Los Angeles, USA, \(\dag \)14 January 2000, Baltimore, USA), mathematician, natural philosopher, and historian of science [9].

  20. 20.

    Eron Lyuttovich Aero (\(^*\)14 May 1934, Naryshkino, Penza district, Soviet Union, \(\dag \)11 July 2016, St. Petersburg, Russia), Soviet/Russian scientist, who made an outstanding contribution to the development of the mechanics of generalized continua such as Cosserat continuum and liquid crystals [35].

  21. 21.

    Vladimir Aleksandrovich Palmov (\(^*\)7 July 1934, Batumi, Soviet Union, \(\dag \)15 October 2018), St. Petersburg, Russian Federation, Soviet/Russian scientist in the field of theoretical and applied mechanics [6, 7].

  22. 22.

    \(^*\)14 March 1879, Ulm, Kingdom of Württemberg, German Empire, \(\dag \)18 April 1955, \(\dag \)18 April 1955, Princeton, New Jersey, US, German-born theoretical physicist.

  23. 23.

    \(^*\)23 January 1862, Königsberg or Wehlau, Prussia, \(\dag \)14 February 1943, Göttingen, Germany, German mathematician.

  24. 24.

    The different interpretations of conservation laws and balance equations will be mentioned here, but not discussed.

  25. 25.

    Johann Peter Gustav Lejeune Dirichlet (\(^*\)13 February 1805, Düren, French Empire, \(\dag \)5 May 1859, Göttingen, Kingdom of Hanover) was a German mathematician.

  26. 26.

    Carl Gottfried Neumann (\(^*\)7 May 1832, Königsberg, Prussia, \(^*\)27 March 1925, Leipzig, Germany) was a German mathematician.

  27. 27.

    Victor Gustave Robin (\(^*\)17 May 1855, Paris, France, \(\dag \)20 November 1897, Paris, France) was a French mathematical analyst and applied mathematician.

  28. 28.

    Pierre Curie (\(^*\)15 May 1859, Paris, France, \(^*\)19 April 1906, Paris, France) was a French physicist, a pioneer in crystallography, magnetism, piezoelectricity, and radioactivity.

  29. 29.

    Franz Ernst Neumann (\(^*\)11 September 1798, Joachimsthal, Holy Roman Empire, \(\dag \)23 May 1895, Königsberg, German Empire) was a German mineralogist and physicist.

  30. 30.

    \(^*\)6 July 1901, Mogilew, Russian Empire, \(\dag \)12 February 1980, Leningrad, Soviet Union, Soviet scientist in the field of theoretical and applied mechanics and control processes.

  31. 31.

    The Russian edition was published in 1970.

  32. 32.

    \(^*\)8 February 1942, Velikiy Ustyug, Vologda region, Soviet Union, \(^*\)4 December 2005, St Petersburg, Russian Federation, Soviet/Russian scientist in the field of theoretical mechanics [11, 21, 22].

  33. 33.

    Around 300 BC, ancient Greek mathematician active as a geometer and logician.

References

  1. Aero, E.L., Kuvshinskii, E.V.: Fundamental equations of the theory of elastic media with rotationally interacting particles. Sov. Phys. Solid State 2(7), 1272–1281 (1961)

    MathSciNet  Google Scholar 

  2. Altenbach, H.: Determination of the reduced properties of multilayer viscoelastic sheets. Mech. Comp. Mater. 24(1), 52–59 (1988). https://doi.org/10.1007/BF00611335

    Article  Google Scholar 

  3. Altenbach, H.: Kontinuumsmechanik - Eine elementare Einführung in die materialunabhängigen und materialabhängigen Gleichungen, 4th edn. Springer (2018). https://doi.org/10.1007/978-3-662-57504-8

  4. Altenbach, H.: Rheological modeling – historical remarks and actual trends in solid mechanics. In: Altenbach, H., Kaplunov, J., Lu, H., Nakada, M. (eds.) Advances in Mechanics of Time-Dependent Materials. Advanced Structured Materials, vol. 188. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-22401-0_1

  5. Altenbach, H., Altenbach, J., Kissing, W.: Mechanics of Composite Structural Elements, 2nd ed. edn. Springer (2018). https://doi.org/10.1007/978-981-10-8935-0

  6. Altenbach, H., Belyaev, A.: Palmov, Vladimir Alexandrovich. In: Altenbach, H., Öchsner , A. (eds.) Encyclopedia of Continuum Mechanics, pp. 1973–1974. Springer Berlin Heidelberg, Berlin, Heidelberg (2020). https://doi.org/10.1007/978-3-662-55771-6_354

  7. Altenbach, H., Belyaev, A., Eremeyev, V.: On the occasion of 75th birthday of Professor Vladimir A. Palmov. ZAMM – J. App. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 89(4), 241 (2009). https://doi.org/10.1002/zamm.200989444

  8. Altenbach, H., Bîrsan, M., Eremeyev, V.A.: Cosserat-type rods. In: Altenbach, H., Eremeyev, V.A. (eds.) Generalized Continua from the Theory to Engineering Applications. CISM International Centre for Mechanical Sciences, vol. 541, pp. 178–248. Springer Vienna, Wien (2013). https://doi.org/10.1007/978-3-7091-1371-4_4

  9. Altenbach, H., Bruhns, O.T.: Truesdell, Clifford Ambrose III. In: Altenbach, H., Öchsner, A.(eds.) Encyclopedia of Continuum Mechanics, pp. 2573–2574. Springer Berlin Heidelberg, Berlin, Heidelberg (2020). https://doi.org/10.1007/978-3-662-55771-6_129

  10. Altenbach, H., Eisenträger, J.: Rheological modeling in solid mechanics from the beginning up to now. Lect. Notes TICMI 22, 13–29 (2021)

    MathSciNet  Google Scholar 

  11. Altenbach, H., Eremeyev, V., Indeitsev, D., Ivanova, E., Krivtsov, A.: On the contributions of Pavel Andreevich Zhilin to mechanics. Technische Mechanik – Eur. J. Eng. Mech. 29(2), 115–134 (2009)

    Google Scholar 

  12. Altenbach, H., Eremeyev, V.A.: Direct approach-based analysis of plates composed of functionally graded materials. Arch. Appl. Mech. 78(10), 775–794 (2008). https://doi.org/10.1007/s00419-007-0192-3

    Article  ADS  Google Scholar 

  13. Altenbach, H., Eremeyev, V.A.: On the analysis of viscoelastic plates made of functionally graded materials. ZAMM – J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 88(5), 332–341 (2008). https://doi.org/10.1002/zamm.200800001

    Article  MathSciNet  ADS  Google Scholar 

  14. Altenbach, H., Eremeyev, V.A.: On the linear theory of micropolar plates. ZAMM – Zeitschrift für Angewandte Mathematik und Mechanik 89(4), 242–256 (2009). https://doi.org/10.1002/zamm.200800207

    Article  MathSciNet  ADS  Google Scholar 

  15. Altenbach, H., Eremeyev, V.A.: Thin-walled structures made of foams. In: Altenbach, H., Öchsner, A. (eds.) Cellular and Porous Materials in Structures and Processes. CISM International Centre for Mechanical Sciences, vol. 521, pp. 167–242. Springer, Wien (2010). https://doi.org/10.1007/978-3-7091-0297-8_4

  16. Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 49(12), 1294–1301 (2011). https://doi.org/10.1016/j.ijengsci.2011.03.011

    Article  MathSciNet  Google Scholar 

  17. Altenbach, H., Eremeyev, V.A.: Cosserat-type shells. In: Altenbach, H., Eremeyev, V.A. (eds.) Generalized Continua from the Theory to Engineering Applications. CISM International Centre for Mechanical Sciences, vol. 541, pp. 131–178. Springer, Wien (2013). DOI https://doi.org/10.1007/978-3-7091-1371-4_3

  18. Altenbach, H., Eremeyev, V.A., Lebedev, L.P., Rendón, L.A.: Acceleration waves and ellipticity in thermoelastic micropolar media. Arch. Appl. Mech. 80(3), 217–227 (2010). https://doi.org/10.1007/s00419-009-0314-1

    Article  ADS  Google Scholar 

  19. Altenbach, H., Eremeyev, V.A., Morozov, N.F.: Surface viscoelasticity and effective properties of thin-walled structures at the nanoscale. Int. J. Eng. Sci. 59, 83–89 (2012). https://doi.org/10.1016/j.ijengsci.2012.03.004

  20. Altenbach, H., Eremeyev, V.A., Morozov, N.F.: On the influence of residual surface stresses on the properties of structures at the nanoscale. In: Altenbach, H., Morozov, N.F. (eds.) Surface Effects in Solid Mechanics, Advanced Structured Materials, vol. 30, pp. 21–32. Springer, Berlin, Heidelberg (2013). https://doi.org/10.1007/978-3-642-35783-1_2

  21. Altenbach, H., Indeitsev, D., Ivanova, E., Krivtsov, A.: In memory of Pavel Andreevich Zhilin (1942–2005). ZAMM – J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 87(2), 79–80 (2007). https://doi.org/10.1002/zamm.200790000

    Article  MathSciNet  ADS  Google Scholar 

  22. Altenbach, H., Ivanova, E.A.: Zhilin, Pavel Andreevich. In: Altenbach, H., Öchsner, A. (eds.) Encyclopedia of Continuum Mechanics, pp. 2792–2795. Springer Berlin Heidelberg, Berlin, Heidelberg (2020). https://doi.org/10.1007/978-3-662-55771-6_147

  23. Altenbach, H., Müller, W.H., Abali, B. (eds.): Higher Gradient Materials and Related Generalized Continua. Advanced Structured Materials, vol. 120. Springer (2019). https://doi.org/10.1007/978-3-030-30406-5

  24. Altenbach, H., Zhilin, P.: Eine allgemeine Theorie elastischer einfacher Schalen (in Russian). Adv. Mech. 11(4), 107–148 (1988)

    Google Scholar 

  25. Altenbach, J., Altenbach, H., Eremeyev, V.A.: On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Arch. App. Mech. 80(1), 73–92 (2010). https://doi.org/10.1007/s00419-009-0365-3

    Article  Google Scholar 

  26. Bellmann, R.: Introduction to Matrix Analysis. Classics in Applied Mathematics, vol. 19, 2nd edn. SIAM, Philadelphia (1997)

    Google Scholar 

  27. Bîrsan, M., Altenbach, H.: On the dynamical theory of thermoelastic simple shells. ZAMM – Zeitschrift für Angewandte Mathematik und Mechanik 91(6), 443–457 (2011). https://doi.org/10.1002/zamm.201000057

  28. Birsan, M., Altenbach, H.: On the Cosserat model for thin rods made of thermoelastic materials with voids. Discrete Continuous Dyn. Syst. – Ser. S 6(6), 1473–1485 (2013). https://doi.org/10.3934/dcdss.2013.6.1473

    Article  MathSciNet  Google Scholar 

  29. Casey, J.: On the derivation of jump conditions in continuum mechanics. Int. J. Struct. Changes Solids 3(2), 61–84 (2011)

    Google Scholar 

  30. Cosserat, E., Cosserat, F.: Théorie des corps déformables. A. Hermann et Fils, Paris (1909)

    Google Scholar 

  31. Curie, P.: Sur la symétrie dans les phénomènes physiques, symétrie d’un champ électrique et d’un champ magnétique. Journal de Physique Théorique et Appliquée 3(1), 393–415 (1894). https://doi.org/10.1051/jphystap:018940030039300

    Article  Google Scholar 

  32. Eremeyev, V.A., Altenbach, H.: On the direct approach in the theory of second gradient plates. In: Altenbach, H., Mikhasev, G.I. (eds.) Shell and Membrane Theories in Mechanics and Biology. Advanced Structured Materials, vol. 45, pp. 147–154. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-02535-3_8

  33. Eremeyev, V.A., Cloud, M.J., Lebedev, L.P.: Applications of Tensor Analysis in Continuum Mechanics. World Scientific, Singapore (2018). https://doi.org/10.1142/10959

  34. Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. SpringerBriefs in Applied Sciences and Technology – Continuum Mechanics. Springer, Berlin, Heidelberg (2013). https://doi.org/10.1007/978-3-642-28353-6

  35. Eremeyev, V.A., Porubov, A.V.: Aero, Eron Lyuttovich. In: Altenbach, H., Öchsner, A. (eds.) Encyclopedia of Continuum Mechanics, pp. 33–35. Springer Berlin Heidelberg, Berlin, Heidelberg (2020). https://doi.org/10.1007/978-3-662-55771-6_27

  36. Ericksen, J.L., Truesdell:, C.: Exact theory of stress and strain in rods and shells. Arch. Ration. Mech. Anal. 1(1), 295–323 (1958). https://doi.org/10.1007/BF00298012

  37. Faddejew, D.K., Faddejewa, W.N.: Numerische Methoden der linearen Algebra. Deutscher Verlag der Wissenschaften, Berlin (1964)

    Google Scholar 

  38. Filin, A.P.: Essays on Mechanical Scientists (in Russian). Strategiya (2007)

    Google Scholar 

  39. Giesekus, H.: Phänomenologische Rheologie: eine Einführung. Springer, Berlin (1994). https://doi.org/10.1007/978-3-642-57953-0

  40. Green, A.E., Naghdi, P.M.: On the derivation of discontinuity conditions in continuum mechanics. Int. J. Eng. Sci. 2(6), 621–624 (1965). https://doi.org/10.1016/0020-7225(65)90040-6

    Article  Google Scholar 

  41. Haupt, P.: Continuum Mechanics and Theory of Materials, 2nd edn. Springer, Berlin (2002)

    Book  Google Scholar 

  42. Haupt, P.: Konzepte der Materialtheorie. Technische Mechanik - European Journal of Engineering Mechanics 16(1), 13–22 (2019)

    Google Scholar 

  43. Hutter, K., Jöhnk, K.: Continuum Methods of Physical Modeling – Continuum Mechanics, Dimensional Analysis, Turbulence. Springer, Berlin, Heidelberg (2004). https://doi.org/10.1007/978-3-662-06402-3

  44. Kolupaev, V.A.: Equivalent Stress Concept for Limit State Analysis. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-73049-3

  45. Kolupaev, V.A., Yu, M.H., Altenbach, H.: Fitting of the strength hypotheses. Acta Mechanica 227(6), 1533–1556 (2016). https://doi.org/10.1007/s00707-016-1566-9

    Article  MathSciNet  Google Scholar 

  46. Krawietz, A.: Materialtheorie - mathematische Beschreibung des phänomenologischen thermomechanischen Verhaltens. Springer, Berlin, Heidelberg, New York and Tokyo (1986). https://doi.org/10.1007/978-3-642-82512-5

  47. Lagally, M.: Vorlesungen über Vektorrechnung. Geest & Portig, Leipzig (1962)

    Google Scholar 

  48. Lemaitre, J., Chaboche, J.L.: Mechanics of Solid Materials. Cambridge University Press, Cambridge (1994). https://doi.org/10.1017/CBO9781139167970

  49. Lurie, A.I.: Non-Linear Theory of Elasticity, North Holland Series in Applied Mathematics and Mechanics, vol. 36. North Holland, Amsterdam (1990)

    Google Scholar 

  50. Lurie, A.I.: Theory of Elasticity. Foundations of Engineering Mechanics. Springer, Berlin-Heidelberg (2005). https://doi.org/10.1007/978-3-540-26455-2

  51. Lurie, S., Volkov-Bogorodskii, D., Altenbach, H., Belov, P., Nazarenko, L.: Coupled problems of gradient thermoelasticity for periodic structures. Arch. Appl. Mech. 93(1), 23–39 (2023). https://doi.org/10.1007/s00419-022-02197-z

    Article  ADS  Google Scholar 

  52. Maugin, G.A.: Continuum Mechanics Through the Eighteenth and Nineteenth Centuries – Historical Perspectives from John Bernoulli (1727) to Ernst Hellinger (1914). Solid Mechanics and Its Applications, vol. 214. Springer, Dordrecht (2013). https://doi.org/10.1007/978-3-319-05374-5

  53. Maugin, G.A.: Continuum Mechanics Through the Twentieth Century – A Concise Historical Perspective, Solid Mechanics and Its Applications, vol. 196. Springer, Dordrecht (2013). https://doi.org/10.1007/978-94-007-6353-1

  54. Müller, W.H., Rickert, W., Vilchevskaya, E.N.: Thence the moment of momentum. ZAMM – J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 100(5), e202000,117 (2020). https://doi.org/10.1002/zamm.202000117

  55. Müller, I.: Thermodynamik: die Grundlagen der Materialtheorie. Bertelsmann Universitätsverlag (1973)

    Google Scholar 

  56. Naumenko, K., Altenbach, H.: Modeling High Temperature Materials Behavior for Structural Analysis – Part I: Continuum Mechanics Foundations and Constitutive Models. Advanced Structured Materials, vol. 28. Springer (2016). https://doi.org/10.1007/978-3-319-31629-1

  57. Nazarenko, L., Glüge, R., Altenbach, H.: Inverse Hooke’s law and complementary strain energy in coupled strain gradient elasticity. ZAMM – J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 101(9), e202100,005 (2021). https://doi.org/10.1002/zamm.202100005

  58. Neumann, F.E.: Vorlesungen über die Theorie der Elastizität der festen Körper und des Lichtäthers (edited by O. E. Meyer). B. G. Teubner-Verlag, Leipzig (1885)

    Google Scholar 

  59. Palmov, V.: Vibrations of Elasto-Plastic Bodies. Springer, Berlin, Heidelberg (1998). https://doi.org/10.1007/978-3-540-69636-0

  60. Pal’mov, V.A.: Fundamental equations of the theory of asymmetric elasticity. J. Appl. Math. Mech. 28(3), 496–505 (1964). https://doi.org/10.1016/0021-8928(64)90092-9

    Article  MathSciNet  Google Scholar 

  61. Palmov, V.A.: Nonlinear Mechanics of Deformable Bodies (in Russian). Polytechnic University Publisher, St. Petersburg (2014)

    Google Scholar 

  62. Pavilainen, G.V., Kuteeva, G.A., Polyakhova, E.N., Rudakova, T.V., Sabaneev, V.S., Tikhonov, A.A.: Essays on the History of Mechanics and Physics (in Russian). VVM, St. Petersburg (2020)

    Google Scholar 

  63. Reiner, M.: Deformation, Strain and Flow: An Elementary Introduction to Rheology. H.K. Lewis, London (1960)

    Book  Google Scholar 

  64. Reiner, M.: Rheologie in elementarer Darstellung. Hanser, München (1967)

    Google Scholar 

  65. Rubin, D., Lai, W.M., Krempl, E.: Introduction to Continuum Mechanics. Butterworth-Heinemann (2010). https://doi.org/10.1016/B978-0-7506-8560-3.X0001-1

    Article  Google Scholar 

  66. Sinkevich, G.I., Nazarov, A.I. (eds.): Mathematical St. Petersburg – History, Science, Sights (in Russian), 2nd edn. Educational Projects, St. Petersburg (2018)

    Google Scholar 

  67. Smol’nikov, B.A.: Mechanics in the History of Science and Society (in Russian). Institute for Computer Research, Moscow, Izhevsk (2013)

    Google Scholar 

  68. Timoshenko, S.P.: History of Strength of Materials. Dover, New York (1983)

    Google Scholar 

  69. Todhunter, I., Pearson, K.: A History of the Theory of Elasticity and of the Strength of Materials from Galilei to the present time, vol. 2: Saint-Venant to Lord Kelvin. Dover, New York (1986)

    Google Scholar 

  70. Todhunter, I., Pearson, K.: A History of the Theory of Elasticity and of the Strength of Materials from Galilei to the present time, vol. 1: Galilei to Saint-Venant 1639–1850. Alpha Editions, New Delhi (2020)

    Google Scholar 

  71. Truesdell, C.: Die Entwicklung des Drallsatzes. ZAMM – J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 44(4–5), 149–158 (1964). https://doi.org/10.1002/zamm.19640440402

    Article  ADS  Google Scholar 

  72. Truesdell, C.: Essays in the History of Mechanics. Springer, Berlin (1968)

    Book  Google Scholar 

  73. Šilhavý, M.: The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin, Heidelberg (1997). https://doi.org/10.1007/978-3-662-03389-0

  74. Wilson, E.B.: Vector Analysis. Founded upon the Lectures of G.W. Gibbs. Yale University Press, New Haven (1901)

    Google Scholar 

  75. Zhilin, P.: Mechanics of deformable directed surfaces. Int. J. Solids Struct. 12, 635–648 (1976). https://doi.org/10.1016/0020-7683(76)90010-X

    Article  MathSciNet  Google Scholar 

  76. Zhilin, P.A.: Vectors and Second Rank Tensors in Three-Dimensional Space (in Russian, Ivanova, E.A., Altenbach, H., Vilchevskaya, E.N., Gavrilov, S.N., Grekova, E.F., Krivtsov, A.M. (eds.)). Nestor, St. Petersburg (2001)

    Google Scholar 

  77. Zhilin, P.A.: Rational Continuum Mechanics (in Russian, Ivanova, E.A., Altenbach, H., Vilchevskaya, E.N., Gavrilov, S.N., Grekova, E.F., Krivtsov, A.M. (eds.)) Polytechnic University Publisher, St. Petersburg (2012)

    Google Scholar 

  78. Zurmühl, R., Falk, S.: Matrizen und ihre Anwendungen, vol. 2. Springer, Berlin (1986)

    Google Scholar 

  79. Zurmühl, R., Falk, S.: Matrizen und ihre Anwendungen, vol. 1. Springer, Berlin (1997)

    Book  Google Scholar 

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Acknowledgements

This contribution is based partly on two books [3, 56]. The last one was written together with Prof. Konstantin Naumenko. The author would like to thank him for his essential contributions in the section.

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    Holm Altenbach

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Correspondence to Holm Altenbach .

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Editors and Affiliations

  1. Institute of Mechanics, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany

    Holm Altenbach

  2. Department of Materials Engineering, Technische Universität München, Munich, Bayern, Germany

    Leonhard Hitzler

  3. Institut für Mechanik, Universität der Bundeswehr München, Neubiberg, Germany

    Michael Johlitz

  4. Institute for Virtual Product Development, Aalen University of Applied Sciences, Aalen, Baden-Württemberg, Germany

    Markus Merkel

  5. Faculty of Mechanical and Systems Engineering, Esslingen University of Applied Sciences, Esslingen am Neckar, Baden-Württemberg, Germany

    Andreas Öchsner

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Altenbach, H. (2024). Continuum Mechanics – Material Independent and Dependent Equations. In: Altenbach, H., Hitzler, L., Johlitz, M., Merkel, M., Öchsner, A. (eds) Lectures Notes on Advanced Structured Materials 2. Advanced Structured Materials, vol 203. Springer, Cham. https://doi.org/10.1007/978-3-031-49043-9_1

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