Theory on the Existence of Solutions in Structural Mechanics

  • Dajun WangEmail author
  • Qishen Wang
  • Beichang (Bert) He


This chapter is devoted to the more fundamental subjects, such as the existence of solutions of static deformation and vibrational modes in the linear theory of Structural Mechanics and the validity of linear theoretical models of structures.


  1. 1.
    Agmon S (1965) Lectures on elliptic boundary value problems. D. Van Nostrand, New YorkGoogle Scholar
  2. 2.
    Benadou M, Ciarlet PG (1975) Sur l’ ellipticite du models linéaire de cogues de W. T. Koiter. Lecture notes in economics and mathematics systems 134, Computing methods in applied science and engineering, 2nd international symposium, Dec 15–19, 1975. Springer, New York, 1976, pp 89–136Google Scholar
  3. 3.
    Benadou M, Lalanne B (1985) Sur l’approximation des coques minces, par des méthods B-splines et éléments finis. In: Grellier JP, Campel GM (eds) Tendances Actuelles en Calcul des structures. Editions Pluralis, Paris, pp 939–958Google Scholar
  4. 4.
    Benadou M, Ciarlet PG, Miara B (1994) Existence theorems for two-dimensional linear shell theories. J Elast 34:111–138CrossRefGoogle Scholar
  5. 5.
    Ciarlet PG, Miara B (1992) Justification of the two-dimensional equations of a linearly elastic shallow shell. Comm Pure Appl Math 45:327–360CrossRefGoogle Scholar
  6. 6.
    Эйдyc Д M (1951) O cмeщaннoй зaдaчe тeopии yпpyгocти. ДAH CCCP, 76(2) (in Russian)Google Scholar
  7. 7.
    Feng K (1979) Elliptic equations on composite manifold and composite elastic structures. Math Num Sinica 1(3):199–208Google Scholar
  8. 8.
    Feng K, Shi ZC (1984) The mathematical theory of elastic structure. Science Press, Beijing, China (in Chinese)Google Scholar
  9. 9.
    Fichera G (1965) Linear elliptic differential systems and eigenvalue problem. Lecture notes in mathematics. Springer, Berlin, Heidelberg, New YorkGoogle Scholar
  10. 10.
    Fichera G (1972) Existence theorems in elasticity. In: Flugge S (ed) Encyclopedia of physics, vol a/2, no 6, pp 347–389CrossRefGoogle Scholar
  11. 11.
    Friedrichs K (1947) On the boundary-value problems of the theory of elasticity and Korn’s inequality. Ann Math 48(2):441–471CrossRefGoogle Scholar
  12. 12.
    Gilbarg D, Trudinger Neil S (1983) Elliptic partial differential equations of second order, 2nd edn. Springer, Berlin, HeidelbergCrossRefGoogle Scholar
  13. 13.
    Gordegiani DG (1974) On the solveabillity of some boundary value problems for a variant of the theory of thin shells. Dokl. Akad, Nauk SSSR, p 215Google Scholar
  14. 14.
    Gurtin ME (1964) Variational principles for linear elastic-dynamics. Arch Ration Mech Anal 16(1):34CrossRefGoogle Scholar
  15. 15.
    Hu HC (1990) Necessary and sufficient condition for correct use of generalized variational principle of elasticity in approximate solutions. Sci China Set A 33(2):196–205Google Scholar
  16. 16.
    Hu HC (1984) Variational principles of theory of elasticity with applications. Science Press, Beijing, China; Gordon, Breath Science Publisher, New YorkGoogle Scholar
  17. 17.
    Korn A (1908) Solution générale du problémé déguilibre dans la théorie de lélasticité dans le cas oń les efforts sont donnés á la surface. Ann. Université ToudouseGoogle Scholar
  18. 18.
    Kreyszig E (1978) Introductory functional analysis with applications. Wiley, New York, Senta Barbare, London, Sydney, TorontoGoogle Scholar
  19. 19.
    Kupradze VD (1965) Potential methods in the theory of elasticity. Israel Program for Scientific Translations, JerusalemGoogle Scholar
  20. 20.
    Kupradze VD (1979) Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. North-Holland Publishing Company, AmsterdamGoogle Scholar
  21. 21.
    Leung AYT, Wang DJ, Wang Q (2004) On concentrated mass and stiffness in structural theories. Int J Struct Stab Dyn 2004(4):171–179Google Scholar
  22. 22.
    Meirovitch L (1980) Computational methods in structural dynamics. SIJTHOFF & NOORDHOFF, Rockville, Maryland, USAGoogle Scholar
  23. 23.
    Mixлин C Г (1950) Пpямыe Meтoды в Maтeмaтичecкoй Физикe. Гocyдapcтвeннoe Издaтeльcтвo Texникo - Teopeтичecкoй Литepaтypы, Mocквa (in Russian)Google Scholar
  24. 24.
    Mixлин C Г (1952) Пpoблeмa Mнимyмa Квaдpaтичнoгo Фyнкциoнaлa. Гocyдap-cтвeннoe Издaтeльcтвo Texникo-Teopeтичecкoй Литepaтypы, Mocквa (in Russian)Google Scholar
  25. 25.
    Y Norio (2008) Oscillation theory of partial differential equations. World Sci Publ, SingaporeGoogle Scholar
  26. 26.
    Payne LE, Weinberger HF (1961) On Korn’s inequality. Arch Ration Mech Anal 8(2):89–98CrossRefGoogle Scholar
  27. 27.
    Shoikhet BA (1974) On existence theorems in linear shell theory. PMM 38:567–571Google Scholar
  28. 28.
    Sun BH (1988) Application and analysis theory of combined elastic structure. Lanzhou University, Lanzhou (in Chinese)Google Scholar
  29. 29.
    Sun BH, Ye ZM (2009) Formulation of elastic multi-structures. Sci China Ser G Phys Mech Astron 52(6):935–953CrossRefGoogle Scholar
  30. 30.
    Sun BH (2012) On existence theorem in theories of elastic structures. Adv Mech 42(5):538–546 (in Chinese)Google Scholar
  31. 31.
    Valid R (1995) The nonlinear theory of shells through variational principles: from elementary algebra to differential geometry. Wiley, New JerseyGoogle Scholar
  32. 32.
    Wang DJ, Hu HC (1982) A unified proof for the positive definiteness and compactness of two kinds of operators in the theory of elastic structure. Acta Mech Sin 14(2):111–121 (in Chinese)Google Scholar
  33. 33.
    Wang DJ, Hu HC (1982) A unified proof of the general properties of the linear vibrations in the theory of elastic structures. J Vib Shock 1(1):6–16 (in Chinese)Google Scholar
  34. 34.
    Wang DJ, Hu HC (1983) A unified proof for the positive-definiteness and compactness of two kinds of operators in the theories of elastic structures. In: Proceedings of the China-France symposium on finite element methods. Science Press China, New York, pp 6–16Google Scholar
  35. 35.
    Wang DJ, Hu HC (1985) Positive definiteness and compactness of two kinds of operators in theory of elastic structures. Scientia Sinica Ser A 28(7):727–739Google Scholar
  36. 36.
    Wang DJ, Wang WQ (1989) The reasonableness problem of theories of structures carrying concentrated masses, springs and supports in vibration problems. Acta Mech Solida Sin 2(2):247–251 (in Chinese)Google Scholar
  37. 37.
    Wang Q, Wang DJ (1993) Singularity under a concentrated force in elasticity. Appl Math Mech 14(8):707–711CrossRefGoogle Scholar
  38. 38.
    Weinberger HF (1974) Variational methods for eigenvalue approximation, 2nd edn, 1987. Society for Industrial and Applied MathematicsGoogle Scholar
  39. 39.
    Wu JK (1981) A proof of ellipticity of the thin shell equations. Acta Mech Solida Sin 2(4):435–444 (in Chinese)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. and Peking University Press 2019

Authors and Affiliations

  1. 1.Department of Mechanics and Engineering SciencePeking UniversityBeijingChina
  2. 2.School of Physics and Electrical EngineeringAnqing Normal UniversityAnqingChina
  3. 3.HBC ConsultingSeattleUSA
  4. 4.State Key Laboratory for Turbulence and Complex SystemsPeking UniversityBeijingChina

Personalised recommendations