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Thermodynamic consistency of beam theories in the context of classical and non-classical continuum mechanics and a thermodynamically consistent new formulation

  • K. S. SuranaEmail author
  • D. Mysore
  • J. N. Reddy
Original Article
  • 37 Downloads

Abstract

In order to enhance currently used beam theories in \(\mathbb {R}^2\) and \(\mathbb {R}^3\) to include mechanisms of dissipation and memory, it is necessary to establish if the mathematical models for these theories can be derived using the conservation and the balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the beam mathematical models. Thermodynamic consistency of the currently used beam models will permit use of entropy inequality to establish constitutive theories in the presence of dissipation and memory mechanism in the currently used beam theories. This is the main motivation for the work presented in this paper. The currently used beam theories are derived based on kinematic assumptions related to the axial and transverse displacement fields. These are then used to derive strain measures followed by constitutive relations. For linear beam theories, strain measures are linear functions of displacement gradients and stresses are linear functions of strain measures. Using these stress and strain measures, energy functional is constructed over the volume of the beam consisting of kinetic energy, strain energy and potential energy of loads. The Euler’s equation(s) extracted from the first variation of this energy functional set to zero yields the differential equations describing the evolution of the deforming beam. Alternatively, principle of virtual work can also be used to derive mathematical models for beams. For linear elastic behavior with small deformation and small strain, the two approaches yield same mathematical models. In this paper we examine whether the currently used beam mathematical models with the corresponding kinematic assumption (i) can be derived using the conservation and balance laws of classical continuum mechanics or (ii) are the conservation and balance laws of non-classical continuum mechanics necessary in their derivation. In order to ensure that the mathematical models for various beam theories result in deformation that is in thermodynamic equilibrium we must establish the consistency of the beam theories with regard to the conservation and the balance laws of continuum mechanics, classical or non-classical in conjunction with their corresponding kinematic assumptions. Currently used Euler–Bernoulli and Timoshenko beam mathematical models that are representative of most beam mathematical models are investigated. This is followed by details of general and higher-order thermodynamically consistent beam theory that is free of kinematic assumptions and other approximations and remains valid for slender as well as deep beams. Model problem studies are presented for slender as well as deep beams. The new formulation presented in this paper ensures thermodynamic equilibrium as it is derived using the conservation and the balance laws of continuum mechanics and remains valid for slender as well as non-slender beams.

Keywords

Beam theories Energy functional Thermodynamic consistency Classical continuum mechanics Non-classical continuum mechanics Internal rotations Cosserat rotations 

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Notes

Acknowledgements

First and third authors are grateful for their endowed professorships and the department of mechanical engineering of the University of Kansas for providing financial support to the second author. Numerical studies presented in this paper as well as the graphical presentation of the results by Mr. Rupesh Anusuri, first author’s graduate student are gratefully acknowledged. The computational facilities provided by the Computational Mechanics Laboratory of the mechanical engineering departments are also acknowledged.

References

  1. 1.
    Ballarini, R.: The Da Vinci–Euler–Bernoulli Beam Theory? https://web.archive.org/web/20060623063248/http://www.memagazine.org/contents/current/webonly/webex418.html. Mechanical Engineering Magazine Online., Archived from the original. http://www.memagazine.org/contents/current/webonly/webex418.html, on 23 July 2006. Retrieved 22 July 2006 (2003)
  2. 2.
    Witmer, E.A.: Elementary Bernoulli–Euler beam theory. MIT Unified Engineering Course Notes, pp. 5–114 to 5–164 (1991–1992)Google Scholar
  3. 3.
  4. 4.
    Han, S.M., Benaroya, H., Wei, T.: Dynamics of Transversely Vibrating Beams using foru Engineering Theories. https://web.archive.org/web/20060623063248/http://www.memagazine.org/contents/current/webonly/webex418.html. Final version. Academic Press, Retrieved 15 Apr 2006 (1999)
  5. 5.
    Timoshenko, S.P.: History of Strength of Materials. McGraw-Hill, New York (1953)Google Scholar
  6. 6.
    Truesdell, C.A.: The rational mechanics of flexible or elastic bodies 1638–1788. Venditioni Exponunt Orell Fussli Turici. Introduction to Vol. X and XI, edn 1. Birkhäuser Basel (1960)Google Scholar
  7. 7.
    Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Dover Publications Inc, New York (2000)zbMATHGoogle Scholar
  8. 8.
    Surana, K.S., Reddy, J.N.: The Finite Element Method for Boundary Value Problems: Mathematics and Computations. CRC/Taylor and Francis, Boca Raton (2017)zbMATHGoogle Scholar
  9. 9.
    Surana, K.S., Reddy, J.N.: The Finite Element Method for Initial Value Problems. CRC/Taylor and Francis, Boca Raton (2017)zbMATHGoogle Scholar
  10. 10.
    Reddy, J.N.: An Introduction to the Finite Element Method, 3rd edn. McGraw-Hill, New York (2006)Google Scholar
  11. 11.
    Levinson, M.: A new rectangular beam theory. J. Sound Vib. 74(1), 81–87 (1981)ADSzbMATHGoogle Scholar
  12. 12.
    Reddy, J.N.: A simple higher-order theory for laminated composite plates. J. Appl. Mech 51(4), 745–752 (1984)ADSzbMATHGoogle Scholar
  13. 13.
    Heyliger, P.R., Reddy, J.N.: A higher-order beam finite element for bending and vibration problem. J. Sound Vib. 126(2), 309–326 (1988)ADSzbMATHGoogle Scholar
  14. 14.
    Hutchinson, J.R.: Shear coefficients for Timoshenko beam theory. J. Appl. Mech. 68(1), 87–98 (2001)ADSzbMATHGoogle Scholar
  15. 15.
    Reddy, J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45(2–8), 288–307 (2007)zbMATHGoogle Scholar
  16. 16.
    Reddy, J.N., Pang, S.D.: Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J. Appl. Phys. 103(2), 023511-1–023511-16 (2008)Google Scholar
  17. 17.
    Reddy, J.N., Arbind, A.: Bending relationship between the modified couple stress-based functionally graded timoshenko beams and homogeneous Bernoulli–Euler beams. Ann. Solid Struc. Mech. 3, 15–26 (2012)Google Scholar
  18. 18.
    Ma, H.M., Gao, X.-L., Reddy, J.N.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56(12), 3379–3391 (2008)ADSMathSciNetzbMATHGoogle Scholar
  19. 19.
    Ma, H.M., Gao, X.-L., Reddy, J.N.: A nonclassical Reddy–Levinson beam model based on a modified couple stress theory. Int. J. Multiscale Comput. Eng. 8(2), 167–180 (2010)Google Scholar
  20. 20.
    Reddy, J.N.: Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int. J. Eng. Sci. 48(11), 1507–1518 (2010)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Reddy, J.N.: Microstructure-dependent couple stress theories of functionally graded beams. J. Mech. Phys. Solids 59(11), 2382–2399 (2011)ADSMathSciNetzbMATHGoogle Scholar
  22. 22.
    Del Piero, G.: A rational approach to Cosserat continua, with application to plate and beam theories. Mech. Res. Commun. 58, 97–104 (2014)Google Scholar
  23. 23.
    Hjelmstad, K.D.: Fundamentals of Structural Mechanics. Prentice Hall, Upper Saddle River (1997)Google Scholar
  24. 24.
    Surana, K.S.: Advanced Mechanics of Continua. CRC/Taylor and Francis, Boca Raton (2015)zbMATHGoogle Scholar
  25. 25.
    Surana, K.S., Powell, M.J., Reddy, J.N.: A more complete thermodynamic framework for solid continua. J. Therm. Eng. 1(1), 1–13 (2015)Google Scholar
  26. 26.
    Surana, K.S., Reddy, J.N., Nunez, D., Powell, M.J.: A polar continuum theory for solid continua. Int. J. Eng. Res. Ind. Appl. 8(2), 77–106 (2015)Google Scholar
  27. 27.
    Surana, K.S., Powell, M.J., Reddy, J.N.: Constitutive theories for internal polar thermoelastic solid continua. J. Pure Appl. Math. Adv. Appl. 14(2), 89–150 (2015)Google Scholar
  28. 28.
    Surana, K.S., Joy, A.D., Reddy, J.N.: A non-classical continuum theory for solids incorporating internal rotations and rotations of Cosserat theories. Contin. Mech. Thermodyn. 29, 665–698 (2017)ADSMathSciNetzbMATHGoogle Scholar
  29. 29.
    Surana, K.S., Joy, A.D., Reddy, J.N.: Non-classical continuum theory and the constitutive theories for thermoviscoelastic solids without memory incorporating internal and cosserat rotations. Acta Mech. (2018) (in print) Google Scholar
  30. 30.
    Surana, K.S., Joy, A.D., Reddy, J.N.: Ordered rate constitutive theories for non-classical thermoviscoelastic solids with memory incorporating internal and Cosserat rotations. Contin. Mech. Thermodyn. (2018).  https://doi.org/10.1007/s00161-018-0697-8
  31. 31.
    Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)zbMATHGoogle Scholar
  32. 32.
    Surana, K.S., Shanbhag, R.S., Reddy, J.N.: Necessity of law of balance of moment of moments in non-classical continuum theories for solid continua. Meccanica 53(11), 2939–2972 (2018)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Prager, W.: Strain hardening under combined stresses. J. Appl. Phys. 16, 837–840 (1945)ADSMathSciNetGoogle Scholar
  34. 34.
    Reiner, M.: A mathematical theory of dilatancy. Am. J. Math. 67, 350–362 (1945)ADSMathSciNetzbMATHGoogle Scholar
  35. 35.
    Todd, J.A.: Ternary quadratic types. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 241, 399–456 (1948)ADSMathSciNetzbMATHGoogle Scholar
  36. 36.
    Rivlin, R.S., Ericksen, J.L.: Stress-deformation relations for isotropic materials. J. Ration. Mech. Anal. 4, 323–425 (1955)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Rivlin, R.S.: Further remarks on the stress-deformation relations for isotropic materials. J. Ration. Mech. Anal. 4, 681–702 (1955)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Wang, C.C.: On representations for isotropic functions, part I. Arch. Ration. Mech. Anal. 33, 249 (1969)Google Scholar
  39. 39.
    Wang, C.C.: On representations for isotropic functions, part II. Arch. Ration. Mech. Anal. 33, 268 (1969)Google Scholar
  40. 40.
    Wang, C.C.: A new representation theorem for isotropic functions, part I and part II. Arch. Ration. Mech. Anal. 36, 166–223 (1970)Google Scholar
  41. 41.
    Wang, C.C.: Corrigendum to ‘representations for isotropic functions’. Arch. Ration. Mech. Anal. 43, 392–395 (1971)Google Scholar
  42. 42.
    Smith, G.F.: On a fundamental error in two papers of C.C. Wang, ‘on representations for isotropic functions, part I and part II’. Arch. Ration. Mech. Anal. 36, 161–165 (1970)Google Scholar
  43. 43.
    Smith, G.F.: On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Int. J. Eng. Sci. 9, 899–916 (1971)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Spencer, A.J.M., Rivlin, R.S.: The theory of matrix polynomials and its application to the mechanics of isotropic continua. Arch. Ration. Mech. Anal. 2, 309–336 (1959)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Spencer, A.J.M., Rivlin, R.S.: Further results in the theory of matrix polynomials. Arch. Ration. Mech. Anal. 4, 214–230 (1960)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Spencer, A.J.M.: Theory of invariants. In: Eringen, A.C. (ed.) Chapter 3 ‘Treatise on Continuum Physics’. Academic Press, New York (1971)Google Scholar
  47. 47.
    Boehler, J.P.: On irreducible representations for isotropic scalar functions. J. Appl. Math. Mech. 57, 323–327 (1977)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Zheng, Q.S.: On the representations for isotropic vector-valued, symmetric tensor-valued and skew-symmetric tensor-valued functions. Int. J. Eng. Sci. 31, 1013–1024 (1993)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Zheng, Q.S.: On transversely isotropic, orthotropic and relatively isotropic functions of symmetric tensors, skew-symmetric tensors, and vectors. Int. J. Eng. Sci. 31, 1399–1453 (1993)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mechanical EngineeringUniversity of KansasLawrenceUSA
  2. 2.Mechanical EngineeringTexas A&M UniversityCollege StationUSA

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