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On the micromechanics theory of Reissner-Mindlin plates

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Summary

A micromechanics model is developed for the Reissner-Mindlin plate. A generalized eigenstrain formulation, i.e., an eigencurvature/eigen-rotation formulation, is proposed, which is the analogue or counterpart of the eigenstrain formulation in linear elasticity. The micromechanics model of the Reissner-Mindlin plate is useful in the study of mechanical behavior of composite plates that contain randomly distributed inhomogeneities, whose sizes are close to the order of thickness of the plate; under those circumstances, the use of micromechanics of linear elasticity is not justified, and moreover, it is inconsistent with structural theories, such as the Reissner-Mindlin plate theory, that are actually used in engineering design.

In this paper, the analytical solution of an elliptical inclusion embedded in an infinite thick plate is sought. In particular, the first order asymptotic (or approximated) solution of the elliptical inclusion problem is obtained in explicit form. Accordingly, the Eshelby tensors of the Reissner-Mindlin plate are derived, which relate eigencurvature and eigen-rotation to the induced curvature and shear deformation fields. Several variational inequalities of the Reissner-Mindlin plate are discussed and derived, including the comparison variational principles of Hashin-Shtrikman/Talbot-Willis, type. As an application, variational bounds are derived to estimate the effective elastic stiffness of Reissner-Mindlin plates, specifically, the flexural rigidity and transverse shear modulus. The newly derived bounds are congruous with the Reissner-Mindlin plate theory, and they provide an optimal estimation on effective rigidity as well as effective transverse shear modulus for unstructured composite thick plates.

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References

  1. Budiansky, B.: On the elastic moduli of some heterogeneous materials. J. Mech. Phy. Solids13, 223–227 (1965).

    Google Scholar 

  2. Caillerie, D.: Non-homogeneous plate theory and conduction in fibered composites. In: Homogenization techniques for composite media (Sanchez-Palencia, E., Zaoui, A., eds.) Berlin: Springer 1986.

    Google Scholar 

  3. Chen, Z.-Q., He, L.-H.: Steady-state response of a Cosserat medium with a spherical inclusion. Acta Mech.116, 97–110 (1996).

    Google Scholar 

  4. Christensen, R. M.: Mechanics of composite materials. New York: Wiley 1979.

    Google Scholar 

  5. Constanda, C.: A mathematical analysis of bending of plates with transverse shear deformation. London: Longman 1990.

    Google Scholar 

  6. Eshelby, J. D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. London, Ser. A,46, 376–396 (1957).

    Google Scholar 

  7. Ekeland, I., Temam, R.: Convex analysis and variational problems. amsterdam: North-Holland 1976.

    Google Scholar 

  8. Green, A. E., Nagdhi, P. M.: The linear theory of an elastic Cosserat plate. Proc. Camb. Phil. Soc.63, 537–550 (1967).

    Google Scholar 

  9. Green, A. E., Naghdi, P. M.: Directed fluid sheets. Proc. R. Soc. Lond, Ser. A,347, 447–473 (1976).

    Google Scholar 

  10. Fleck, N., Hutchinson, J. W.: Strain gradient platicity. Adv. Appl. Mech.33, 295–361 (1995).

    Google Scholar 

  11. Green, A. E., Naghdi, P. M.: A direct theory of viscous fluid flow in pipes I. Basic general developments. Phil. Trans. R. Soc. Lond. A342, 525–542 (1993).

    Google Scholar 

  12. Green, A. E., Naghdi, P. M.: A direct theory of viscous fluid flow in pipes II. Flow of incompressible viscous fluid in curved pipes. Phil. Trans. R. Soc. Lond. A342, 543–572 (1993).

    Google Scholar 

  13. Hashin, Z., Shtrikman, S.: On some variational principles in anisotropic and nonhomogeneous elasticity. J. Mech. Phys. Solids10, 335–342 (1962).

    Google Scholar 

  14. Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of polycrystals. J. Mech. Phys. Solids10, 343–352 (1962).

    Google Scholar 

  15. Hashin, Z.: Theory of mechanical behaviour of heterogeneous materials. Appl. Mech. Rev.17, 1–9 (1964).

    Google Scholar 

  16. Hill, R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids11, 357–372 (1963).

    Google Scholar 

  17. Hill, R.: New derivations of some elastic extremum principles. In: Progress in applied mechanics—The Prager anniversary volume, pp. 99–106. New York: Macmillan 1963.

    Google Scholar 

  18. Hill, R.: Theory of mechanical properties of fiber-strengthened materials-III: self-consistent model. J. Mech. Phys. Solids13, 189–198 (1965).

    Google Scholar 

  19. Hill, R.: A self-consistent mechanics of composite materials. J. Mech. Phys. Solids13, 213–222 (1965).

    Google Scholar 

  20. Kaprielion, P. V., Rogers, T. G., Spencer, A. J. M.: Theory of laminated elastic plates I. Isotropic laminae. Phil. Trans. R. Soc. Lond. A324, 565–594 (1988).

    Google Scholar 

  21. Karam, V. J., Telles, J. C. F.: On boundary elements for Reissner's plate theory. Eng. Anal.5, 21–27 (1988).

    Google Scholar 

  22. Kröner, E.: Linear properties of random media: the systematic theory. In: Rheological behaviour and structure of materials (Huet, C., Zaoui, A., eds.), pp. 14–50. Paris: Press ENPC 1981.

    Google Scholar 

  23. Kröner, E.: Statistical modelling. In: Modelling small deformation of polycrystals (Gittus, J., Zarka, J., eds.), pp. 229–291. New York: Elsevier 1986.

    Google Scholar 

  24. Kellog, O. D.: Foundations of potential theory. New York: Dover 1953.

    Google Scholar 

  25. Love, A. E. H.: A treatise on the mathematical theory of elasticity. Cambridge: Cambridge University Press 1926.

    Google Scholar 

  26. Li, S.: The micromechanics theory of classical plates: congruous estimate of overall elastic stiffness. Int. J. Solids Struct. (in press).

  27. Mindlin, R. D.: Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. ASME J. Appl. Mech.18, 31–38 (1951).

    Google Scholar 

  28. Mindlin, R. D.: On Reissner's equations for sandwich plates. In: Mechanics today (Nemat-Nasser, S., ed.), Vol. 5, pp. 315–328. New York: Pergman Press 1980.

    Google Scholar 

  29. Mura, T.: Micromechanics of defects in solids. Dordrecht: Martinus Nijhoff 1987.

    Google Scholar 

  30. Naghdi, P. M.: Foundation of elastic shell theory. In: Progress in solid mechanics, vol. 4 (Sneddon, I. N., Hill, R., eds.). Amsterdam, North-Holland 1963.

    Google Scholar 

  31. Naghdi, P. M.: The theory of shells and plates. In: Handbuch der Physik: Mechanics of solids II, vol. VI a/2 (Flügge, S., Truesdell, C., eds.). Berlin: Springer 1972.

    Google Scholar 

  32. Nemat-Nasser, S., Hori, M.: Micromechanics: overall properties of heterogeneous materials. Amsterdam: North-Holland 1993.

    Google Scholar 

  33. Parton, V. Z., Kudryavtsev, B. A.: Engineering mechanics of composite structures. Boca Raton: CRC Press 1993.

    Google Scholar 

  34. Qin, S., Fan, H., Mura, T.: The eigenstrain formulation for classical plates. Int. J. Solids Struct.28, 363–372 (1991).

    Google Scholar 

  35. Reissner, E.: On the theory of bending of elastic plates. J. Math. Phys.23, 184–191 (1944).

    Google Scholar 

  36. Reissner, E.: The effect of transverse shear deformation on the bending of elastic plate. ASME J. Appl. Mech.12, A69-A77 (1945).

    Google Scholar 

  37. Reissner, E.: On bending of elastic plates. Quart. J. Appl. Math.5, 268–278 (1947).

    Google Scholar 

  38. Talbot, D. R. S., Willis, J. R.: Variational principles for inhomogeneous non-linear media. IMA J. Appl. Math.35, 39–54 (1985).

    Google Scholar 

  39. Tanaka, K., Mori, T.: Note on volume integrals of the elastic field around an ellipsoidal inclusion. J. Elasticity2, 199–200 (1972).

    Google Scholar 

  40. Torquato, S.: Effective stiffness tensor of composite media-I exact series expansions. J. Mech. Phys. Solids45, 1421–1448 (1997).

    Google Scholar 

  41. Vander Weeën, F.: Application of the boundary integral equation method to Reissner's plate model. Int. J. Num. Methods Eng.18, 1–10 (1982).

    Google Scholar 

  42. Walpole, L. J.: On bounds for the overall elastic moduli of inhomogeneous systems-I. J. Mech. Phys. Solids14, 151–162 (1966).

    Google Scholar 

  43. Walpole, L. J.: On bounds for the overall elastic moduli of inhomogeneous systems-II. J. Mech. Phys. Solids14, 289–301 (1966).

    Google Scholar 

  44. Walpole, L. J.: On the overall elastic moduli of composite materials. J. Mech. Phys. Solids17, 235–251 (1969).

    Google Scholar 

  45. Walpole, L. J.: Elastic behavior of composite materials: theoretical foundations. In Adv. Appl. Mech.21, 169–242 (Yih, C.-S., ed.). New York: Academic Press 1981.

    Google Scholar 

  46. Walpole, L. J.: A translated rigid ellipsoidal inclusion in an elastic medium. Proc. R. Soc. Lond. A434, 571–585 (1991).

    Google Scholar 

  47. Watson, G. N.: Theory of Bessel functions, 2nd ed. Cambridge: Cambridge University Press 1944.

    Google Scholar 

  48. Willis, J. R.: Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solids25, 185–202 (1977).

    Google Scholar 

  49. Willis, J. R.: Variational and related methods for the overall properties of composites. In: Advances in applied mechanics21, pp. 1–78 (Yih, C.-S., ed.). New York: Academic Press 1981.

    Google Scholar 

  50. Willis, J. R.: The structure of overall constitutive relations for a class of nonlinear composites. IMA J. Appl. Math.43, 231–242 (1989).

    Google Scholar 

  51. Willis, J. R.: Bounds for the properties of nonlinear composites. Lecture notes for the course offered in Institute for Mechanics and Materials: The Mechanics-Materials Linkage 1995 Summer School, Northwestern University 1995.

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  1. S. Li

    Present address: Department of Mechanical Engineering, McCormick School of Engineering and Applied Science, Northwestern University, IL 60208, Evanston, Illinois, USA

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    Li, S. On the micromechanics theory of Reissner-Mindlin plates. Acta Mechanica 142, 47–99 (2000). https://doi.org/10.1007/BF01190012

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