Skip to main content
Log in

Experimental tests of rotation sensitivity in Cosserat elasticity and in gravitation

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

Experimental tests of rotation sensitivity in elastic materials and in empty space are surveyed. Pioneers in the theory for each field of study were aware of the other field. Sensitivity to rotation has been demonstrated experimentally in elastic solids but tests for rotation gradient sensitivity in gravity have found no such effects. Insight from comparison of experimental approaches may lead to new experimental protocols.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Einstein, A.: Zür allgemeinen Relativitätstheorie, Preuss. Akad. Wiss. Berlin, Sitzber. 778-786, (1915)

  2. Einstein, A.: Zür allgemeinen Relativitätstheorie (Nachtrag), Preuss. Akad. Wiss. Berlin, Sitzber., 799-801, (1915)

  3. Cartan, É.: Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion, C. R. Acad. Sci. (Paris) 174 593-595 (1922), English translation by G. D. Kerlick, On a generalization of the notion of Riemann curvature and spaces with torsion. In: Proc. of the 6th Course of Internat. School on Cosmology and Gravitation: Spin, Torsion, Rotation, and Supergravity, (Erice, 1979) Eds. P. G. Bergmann and V. De Sabbata 489-491 (Plenum: New York, 1980)

  4. Cosserat, E., Cosserat, F.: Théorie des Corps Déformables. Hermann et Fils, Paris (1909)

    MATH  Google Scholar 

  5. Hehl, F.W.: Spin and torsion in general relativity: I. Foundations, General relativity and gravitation 4(4), 333–349 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hehl, F.W., von der Heyde, P., Kerrlick, G.D., Nester, J.M.: General relativity with spin and torsion: Foundations and prospects. Rev. Mod. Phys. 48, 393–416 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hammond, R.T.: Torsion gravity. Rep. Prog. Phys. 65, 599–649 (2002)

    Article  MathSciNet  Google Scholar 

  8. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W. H. Freeman, San Francisco (1973)

    Google Scholar 

  9. Abbott, B.P., et al.: Observation of Gravitational Waves from a Binary Black Hole Merger. Phys. Rev. Lett. 116, 061102 (2016)

    Article  MathSciNet  Google Scholar 

  10. Abbott, B.P., et al.: GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral. Phys. Rev. Lett. 119, 161101 (2017)

    Article  Google Scholar 

  11. Everitt, C.W.F., et al.: Gravity Probe B: Final Results of a Space Experiment to Test General Relativity. Phys. Rev. Lett. 106, 221101 (2011)

    Article  Google Scholar 

  12. Hojman, S., Rosenbaum, M., Ryan, M.P.: Gauge invariance, minimal coupling, and torsion. Phys. Rev. D, 13141–13146, (1978)

  13. Trautman, A.: Einstein-Cartan Theory, Encyclopedia of Mathematical Physics, Francoise, J.P., Naber, G.L., Tsou, S.T. (eds.), Oxford: Elsevier, vol. 2, pages 189-195 (2006). [arXiv:0711.1535]

  14. Finsler, P.: Über Kurven und Flächen in allgemeinen Räumen, Dissertation, Gottingen, (1918). Birkhauser Verlag, Basel (1951)

  15. Ishikawa, H.: Note on Finslerian relativity. Journal of Mathematical Physics 22, 995–1004 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  16. Basilakos, S., Kouretsis, A.P., Saridakis, E.N., Stavrinos, P.C.: Resembling dark energy and modified gravity with Finsler-Randers cosmology. Phys. Rev. D 88, 123510 (2013)

    Article  Google Scholar 

  17. Ni, W.T.: Hojman-Rosenbaum-Ryan-Shepley torsion theory and Eötvös-Dicke-Braginsky experiments. Phys. Rev. D, 192260–3, (1979)

  18. Roll, P.G., Krotkov, R., Dicke, B.H.: The equivalence of inertial and passive gravitational mass. Annals of Physics, New York 26, 442–517 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  19. Braginsky, V.H., Panov, V.I.: Verification of the Equivalence of Inertial and Gravitational Mass. Zh. Eksp. Yeor. Fiz. 61, 873 (1971)

    Google Scholar 

  20. Braginsky, V.H., Panov, V.I.: Verification of the Equivalence of Inertial and Gravitational Mass. Sov. Phys. JETP 34, 463 (1972)

    Google Scholar 

  21. Eötvös, R.V.: Über die Anziehung der Erde auf verschiedene Substanzen. Math Naturw. Ber. aus Ungarn 8, 65–68 (1889)

    Google Scholar 

  22. Eötvös, R.V., Pekar, D., Fekete, E.: Beitragezum Gezetze der Proportionalität von Trägheit und Gravität. Annalen der Physik 68, 11–66 (1922)

    Article  Google Scholar 

  23. Ritter, R.C., Goldblum, C.E., Ni, W.T., Gillies, G.T., Speake, C.C.: Experimental test of equivalence principle with polarized masses. Phys. Rev. D 42, 977–991 (1979)

    Article  Google Scholar 

  24. Ritter, R.C., Winkler, L.I., Gillies, G.T.: Search for Anomalous Spin-Dependent Forces with a Polarized-Mass Torsion Pendulum. Phys. Rev. Lett. 70, 701–705 (1993)

    Article  Google Scholar 

  25. Chui, T.C.P., Ni, W.: Experimental search for an anomalous spin-spin interaction between electrons. Phys. Rev. Lett. 71, 3247–50 (1993)

    Article  Google Scholar 

  26. Mao, Y., Tegmark, M., Guth, A.H., Cabi, S.: Constraining torsion with Gravity Probe B. Phys. Rev. D 76, 104029 (2007)

    Article  Google Scholar 

  27. March, R., Bellettini, G., Tauraso, R., Dell’Agnello, S.: Constraining spacetime torsion with the Moon and Mercury. Phys. Rev. D 83, 104008 (2011)

    Article  MATH  Google Scholar 

  28. Kostelecký, V.A., Russell, N., Tasson, J.D.: Constraints on Torsion from Bounds on Lorentz Violation. Phys. Rev. Lett. 100, 111102 (2008)

    Article  Google Scholar 

  29. Heckel, B.R., Terrano, W.A., Adelberger, E.G.: Limits on Exotic Long-Range Spin-Spin Interactions of Electrons. Phys. Rev. Lett. 111, 151802 (2013)

    Article  Google Scholar 

  30. E. Whittaker, A history of the theories of aether and electricity, Dover, New York, 1989, reprint of London: Thomas Nelson and Sons, (1951)

  31. Kelvin, William Thomson Lord: XLVI. On the reflexion and refraction of light. Philosophical Magazine 26, 414–425 (1888)

    Google Scholar 

  32. Thomson, William, Kelvin, Baron: Mathematical and Physical Papers: 1841-1890 Volume 3, Cambridge University Press; J. J. Clay and sons, University Press, London, 1890. XLVI, On a gyrostatic adynamic constitution for ‘ether’, p.466-472, reprinted from Edin. Roy. Soc. Proc. Vol. xvii. , pp. 127-132 March 17, (1890)

  33. Maxwell, J.C.: The scientific papers of James Clerk Maxwell, ed. Cambridge University Press, Cambridge, W. D. Niven (1890), reprinted by Dover, New York, (1952)

  34. FitzGerald, G.: The scientific writings of the late George Francis FitzGerald, ed. Joseph Larmor, Hodges and Figgis, Dublin (1902)

  35. Hunt, B.J.: The Maxwellians. Cornell University Press, Ithaca, New York and London (1991)

    Google Scholar 

  36. MacCullagh, J.: An essay towards the dynamical theory of crystalline reflexion and refraction (read 9 Dec. 1839). Trans. Royal Irish Acad. Sci. 21, 17–50 (1848)

    Google Scholar 

  37. Sokolnikoff, I.S.: Mathematical Theory of Elasticity. Krieger, Malabar, FL (1983)

    MATH  Google Scholar 

  38. Milton, G.W., Cherkaev, A.V.: Which elasticity tensors are realizable? ASME J. Eng. Mater. Technol. 117, 483–493 (1995)

    Article  Google Scholar 

  39. Zhu, H.X., Knott, J.F., Mills, N.J.: Analysis of the elastic properties of open-cell foams with tetrakaidecahedral cells. J. Mech. Phys. Solids 45, 319–343 (1997)

    Article  Google Scholar 

  40. Lakes, R.S.: Foam structures with a negative Poisson’s ratio. Science 235, 1038–1040 (1987)

    Article  Google Scholar 

  41. Grima, J.N., Evans, K.E.: Auxetic behavior from rotating squares. J. Mater. Sci. Lett. 19, 1563–1565 (2000)

    Article  Google Scholar 

  42. Andrade, C., Ha, C.S., Lakes, R.S.: Extreme Cosserat elastic cube structure with large magnitude of negative Poisson’s ratio. Journal of Mechanics of Materials and Structures (JoMMS) 13(1), 93–101 (2018)

    Article  MathSciNet  Google Scholar 

  43. Lakes, R.S., Lee, T., Bersie, A., Wang, Y.C.: Extreme damping in composite materials with negative stiffness inclusions. Nature 410, 565–567 (2001)

    Article  Google Scholar 

  44. Lakes, R.S.: Extreme damping in compliant composites with a negative stiffness phase. Philosophical Magazine Letters 81, 95–100 (2001)

    Article  Google Scholar 

  45. Jaglinski, T., Stone, D.S., Kochmann, D., Lakes, R.S.: Materials with viscoelastic stiffness greater than diamond. Science 315, 620–622 (2007)

    Article  Google Scholar 

  46. Moore, B., Jaglinski, T., Stone, D.S., Lakes, R.S.: Negative incremental bulk modulus in foams. Philosophical Magazine Letters 86, 651–659 (2006)

    Article  Google Scholar 

  47. Mindlin, R.D.: Stress functions for a Cosserat continuum. Int. J. Solids Structures 1, 265–271 (1965)

    Article  Google Scholar 

  48. Eringen, A.C.: Theory of micropolar elasticity. In: Fracture,1, 621-729, Liebowitz, H. (ed.), Academic Press, New York, (1968)

  49. Koiter, W.T.: Couple-Stresses in the theory of elasticity, Parts I and II. Proc. Koninklijke Ned. Akad. Wetenshappen 67, 17–44 (1964)

    MathSciNet  MATH  Google Scholar 

  50. Berglund, K.: Investigation of a two dimensional model of a micropolar continuum. Archives of Mechanics (Warsaw) 29(3), 383–392 (1977)

    MATH  Google Scholar 

  51. Berglund, K.: Mechanics of Micropolar Media. In: Brulin, O., Hsieh, R. (eds.) Structural models of micropolar media. World Scientific, Singapore (1982)

    Chapter  Google Scholar 

  52. Hehl, F.W., Obukhov, Y.N.: Élie Cartan’s torsion in geometry and in field theory, an essay. Ann. Fond L. de Broglie 32, 157 (2007). (arXiv:gr-qc/0606062)

    MATH  Google Scholar 

  53. Gauthier, R.D., Jahsman, W.E.: A quest for micropolar elastic constants. J. Applied Mechanics 42, 369–374 (1975)

    Article  MATH  Google Scholar 

  54. Bigoni, D., Drugan, W.J.: Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. J. Appl. Mech. 74, 741–753 (2007)

    Article  MathSciNet  Google Scholar 

  55. Schijve, J.: Note on couple stresses. J. Mech. Phys. Solids 14, 113–120 (1966)

    Article  Google Scholar 

  56. Ellis, R.W., Smith, C.W.: A thin plate analysis and experimental evaluation of couple stress effects. Exp. Mech 7, 372–380 (1968)

    Article  Google Scholar 

  57. Yang, J.F.C., Lakes, R.S.: Transient study of couple stress in compact bone: torsion. Journal of Biomechanical Engineering 103, 275–279 (1981)

    Article  Google Scholar 

  58. Lakes, R.S.: Experimental microelasticity of two porous solids. International Journal of Solids and Structures 22, 55–63 (1986)

    Article  Google Scholar 

  59. Rueger, Z., Lakes, R.S.: Experimental study of elastic constants of a dense foam with weak Cosserat coupling. J. Elasticity 137, 101–115 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  60. Merkel, A., Tournat, V.: Experimental Evidence of Rotational Elastic Waves in Granular Phononic Crystals. Phys. Rev. Lett. 107, 225502 (2011)

    Article  Google Scholar 

  61. Lakes, R.S., Gorman, D., Bonfield, W.: Holographic screening method for microelastic solids. J. Materials Science 20, 2882–2888 (1985)

    Article  Google Scholar 

  62. Park, H.C., Lakes, R.S.: Cosserat micromechanics of human bone: strain redistribution by a hydration-sensitive constituent. J. Biomechanics 19, 385–397 (1986)

    Article  Google Scholar 

  63. Lakes, R.S.: Reduced warp in torsion of reticulated foam due to Cosserat elasticity: experiment. Zeitschrift fuer Angewandte Mathematik und Physik (ZAMP) 67(3), 1–6 (2016)

    MathSciNet  MATH  Google Scholar 

  64. Rueger, Z., Lakes, R.S.: Strong Cosserat elasticity in a transversely isotropic polymer lattice. Phys. Rev. Lett. 120, 065501 (2018)

    Article  Google Scholar 

  65. Rueger, Z., Lakes, R.S.: Experimental Cosserat elasticity in open cell polymer foam. Philosophical Magazine 96, 93–111 (2016)

    Article  Google Scholar 

  66. Rueger, Z., Lakes, R.S.: Strong Cosserat elastic effects in a unidirectional composite, Zeitschrift für angewandte Mathematik und Physik (ZAMP) 68:54, 9 pages (2017)

  67. Ha, C.S., Plesha, M.E., Lakes, R.S.: Chiral three-dimensional isotropic lattices with negative Poisson’s ratio. Physica Status Solidi B 253, 1243–1251 (2016)

    Article  Google Scholar 

  68. Reasa, D.R., Lakes, R.S.: Nonclassical Chiral Elasticity of the Gyroid Lattice. Phys. Rev. Lett. 125(20), 205502 (2020)

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge support of this research by the National Science Foundation via Grant No. CMMI-1906890.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to R. S. Lakes.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lakes, R.S. Experimental tests of rotation sensitivity in Cosserat elasticity and in gravitation. Z. Angew. Math. Phys. 72, 131 (2021). https://doi.org/10.1007/s00033-021-01563-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-021-01563-1

Keywords

Mathematics Subject Classification

Navigation