171
Views
0
CrossRef citations to date
0
Altmetric
Original Articles

A finite deformation, finite strain nonclassical internal polar continuum theory for solids

, &
Pages 381-393 | Received 25 Jan 2017, Accepted 06 Jun 2017, Published online: 17 Nov 2017

References

  • K. S. Surana, M. J. Powell, and J. N. Reddy, “A more complete thermodynamic framework for solid continua,” J. Ther. Eng., vol. 1, no. 1, pp. 1–13, 2015.
  • K. S. Surana, J. N. Reddy, D. Nunez, and M. J. Powell, “A polar continuum theory for solid continua,” Int. J. Eng. Res. Indus. Appli., vol. 8, no. 2, pp. 77–106, 2015.
  • A. C. Eringen. “Mechanics of Micromorphic Materials.“ Proc. 11th Intern. Congress. Appl. Mech., H. Gortler (Ed.), Berlin: Springer-Verlag, 1964a, pp. 131–138.
  • A. C. Eringen. “Mechanics of Micromorphic Continua.” Mechanics of Generalized Continua, E. Kroner (Ed.), Berlin: Springer-Verlag, 1968, pp. 18–35.
  • A. C. Eringen. “Theory of micropolar elasticity.” Fracture, H. Liebowitz (Ed.), NY: Academic Press, 1968, pp. 621–729.
  • A. C. Eringen, “Balance laws of micromorphic mechanics,” Int. J. Eng. Sci., vol. 8, no. 10, pp. 819–828, 1970.
  • A. C. Eringen, “Theory of thermo-microstretch fluids and bubbly liquids,” Int. J. Eng. Sci., vol. 28, no. 2, pp. 133–143, 1990.
  • A. C. Eringen, Theory of Micropolar Elasticity, New York, NY, USA: Springer-Verlag, 1990.
  • A. C. Eringen, “A unified theory of thermomechanical materials,” Int. J. Eng. Sci., vol. 4, pp. 179–202, 1966.
  • A. C. Eringen, “Linear theory of micropolar viscoelasticity,” Int. J. Eng. Sci., vol. 5, pp. 191–204, 1967.
  • A. C. Eringen, “Theory of micromorphic materials with memory,” Int. J. Eng. Sci., vol. 10, pp. 623–641, 1972.
  • W. Koiter. “Couple stresses in the theory of elasticity, I and II,” Nederl. Akad. Wetensch. Proc. Ser. B, vol. 67, pp. 17–44, 1964.
  • W. Oevel and J. Schröter, “Balance equations for micromorphic materials,” J. Stat. Phys., vol. 25, no. 4, pp. 645–662, 1981.
  • J. N. Reddy, “Nonlocal theories for bending, buckling and vibration of beams,” Int. J. Eng. Sci., vol. 45, pp. 288–307, 2007.
  • J. N. Reddy and S. D. Pang, “Nonlocal continuum theories of beams for the analysis of carbon nanotubes,” J. Appl. Phys., vol. 103, p. 023511, 2008.
  • J. N. Reddy, “Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates,” Int. J. Eng. Sci., vol. 48, pp. 1507–1518, 2010.
  • P. Lu, P. Q. Zhang, H. P. Leo, C. M. Wang, and J. N. Reddy, “Nonlocal elastic plate theories,” Proc. Roy. Soc. A, vol. 463, pp. 3225–3240, 2007.
  • J. F. C. Yang and R. S. Lakes, “Experimental study of micropolar and couple stress elasticity in compact bone in bending,” J. Biomech., vol. 15, no. 2, pp. 91–98, 1982.
  • V. A. Lubarda and X. Markenscoff, “Conservation integrals in couple stress elasticity,” J. Mech. Phys. Solids, vol. 48, pp. 553–564, 2000.
  • H. M. Ma, X. L. Gao, and J. N. Reddy, “A microstructure-dependent timoshenko beam model based on a modified couple stress theory,” J. Mech. Phys. Solids, vol. 56, pp. 3379–3391, 2008.
  • H. M. Ma, X. L. Gao, and J. N. Reddy, “A nonclassical reddy-levinson beam model based on modified couple stress theory,” J. Multiscale Comput. Eng., vol. 8, no. 2, pp. 167–180, 2010.
  • J. N. Reddy, “Microstructure dependent couple stress theories of functionally graded beams,” J. Mech. Phys. Solids, vol. 59, pp. 2382–2399, 2011.
  • J. N. Reddy and A. Arbind, “Bending relationship between the modified couple stress-based functionally graded timoshenko beams and homogeneous Bernoulli-Euler beams,” Ann. Solid Struc. Mech., vol. 3, pp. 15–26, 2012.
  • A. R. Srinivasa and J. N. Reddy, “A model for a constrained, finitely deforming elastic solid with rotation gradient dependent strain energy and its specialization to Von Kármán plates and beams,” J. Mech. Phys. Solids, vol. 61, no. 3, pp. 873–885, 2013.
  • R. J. Mora and A. M. Waas, “Evaluation of the micropolar elasticity constants for honeycombs,” Acta Mech., vol. 192, pp. 1–16, 2007.
  • P. R. Onck, “Cosserat modeling of cellular solids,” C. R. Mecanique, vol. 330, pp. 717–722, 2002.
  • P. H. Segerstad, S. Toll, and R. Larsson, “A micropolar theory for the finite elasticity of open-cell cellular solids,” Proc. Roy. Soc. A, vol. 465, pp. 843–865, 2009.
  • H. Altenbach and V. A. Eremeyev, “On the linear theory of micropolar plates,” ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, vol. 89, no. 4, pp. 242–256, 2009.
  • H. Altenbach and V. A. Eremeyev, “Strain rate tensors and constitutive equations of inelastic micropolar materials,” Int. J. Plast., vol. 63, pp. 3–17, 2014.
  • H. Altenbach, V. A. Eremeyev, L. P. Lebedev, and L. A. Rendón, “Acceleration waves and ellipticity in thermoelastic micropolar media,” Arch. Appl. Mech., vol. 80, no. 3, pp. 217–227, 2010.
  • H. Altenbach, G. A. Maugin, and V. Erofeev, Mechanics of Generalized Continua, vol. 7, New York, NY, USA: Springer-Verlag, 2011.
  • H. Altenbach, K. Naumenko, and P. A. Zhilin, “A micro-polar theory for binary media with application to phase-transitional flow of fiber suspensions,” Cont. Mech. Thermody., vol. 15, no. 6, pp. 539–570, 2003.
  • F. Ebert, “A similarity solution for the boundary layer flow of a polar fluid,” Chem. Eng. J., vol. 5, no. 1, pp. 85–92, 1973.
  • V. A. Eremeyev, L. P. Lebedev, and H. Altenbach, “Kinematics of micropolar continuum,” in Foundations of Micropolar Mechanics, New York, NY, USA: Springer-Verlag, 2013, pp. 11–13.
  • V. A. Eremeyev and W. Pietraszkiewicz, “Material symmetry group and constitutive equations of anisotropic Cosserat continuum,” Generalized Continua As Models for Materials. Berlin: Springer-Verlag, 2012, p. 10.
  • V. A. Eremeyev and W. Pietraszkiewicz, “Material symmetry group of the non-linear polar-elastic continuum,” Int. J. Solids Struct., vol. 49, no. 14, pp. 1993–2005, 2012.
  • E. F. Grekova, “Ferromagnets and Kelvin’s medium: Basic equations and wave processes,” J. Comput. Acou., vol. 9, no. 02, pp. 427–446, 2001.
  • E. F. Grekova, “Linear reduced Cosserat medium with spherical tensor of inertia, where rotations are not observed in experiment,” Mech. Solids, vol. 47, no. 5, pp. 538–543, 2012.
  • E. F. Grekova, M. A. Kulesh, and G. C. Herman, “Waves in linear elastic media with microrotations, part 2: Isotropic reduced Cosserat model,” Bull. Seismol. Soc. Am., 99(2B):1423–1428, 2009.
  • G. Grioli, “Linear micropolar media with constrained rotations,” in Micropolar Elasticity, New York, NY, USA: Springer-Verlag, 1974, pp 45–71.
  • E. F. Grekova and G. A. Maugin, “Modelling of complex elastic crystals by means of multi-spin micromorphic media,” Int. J. Eng. Sci., vol. 43, no. 5, pp. 494–519, 2005.
  • G. Grioli, “Microstructures as a refinement of cauchy theory. problems of physical concreteness,” Cont. Mech. Thermodyn., vol. 15, no. 5, pp. 441–450, 2003.
  • J. Altenbach, H. Altenbach, and V. A. Eremeyev, “On generalized Cosserat-type theories of plates and shells: a short review and bibliography,” Arch. Appl. Mech., vol. 80, no. 1, pp. 73–92, 2010.
  • M. Lazar and G. A. Maugin, “Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity,” Int. J. Eng. Sci., vol. 43, no. 13, pp. 1157–1184, 2005.
  • M. Lazar and G. A. Maugin, “Defects in gradient micropolar elasticity I: screw dislocation,” J. Mech. Phys. Solids, vol. 52, no. 10, pp. 2263–2284, 2004.
  • G. A. Maugin, “A phenomenological theory of ferroliquids,” Int. J. Eng. Sci., vol. 16, no. 12, pp. 1029–1044, 1978.
  • G. A. Maugin, “Wave motion in magnetizable deformable solids,” Int. J. Eng. Sci., vol. 19, no. 3, pp. 321–388, 1981.
  • G. A. Maugin, “On the structure of the theory of polar elasticity,” Philosoph. Transac. Roy Soc. London A Math. Phys. Eng. Sci., vol. 356, no. 1741, pp. 1367–1395, 1998.
  • W. Pietraszkiewicz and V. A. Eremeyev, “On natural strain measures of the non-linear micropolar continuum,” Int. J. Solids Struct., vol. 46, no. 3, pp. 774–787, 2009.
  • E. Cosserat and F. Cosserat, “Théorie des corps déformables” by E. and F. Cosserat, publisher A. Hermann et fils, Paris. 1909.
  • A. V. Zakharov and E. L. Aero, “Statistical mechanical theory of polar fluids for all densities,” Phys. A Stat. Mech. Appl., vol. 160, no. 2, pp. 157–165, 1989.
  • A. E. Green, “Micro-materials and multipolar continuum mechanics,” Int. J. Eng. Sci., vol. 3, no. 5, pp. 533–537, 1965.
  • A. E. Green and R. S. Rivlin, “The relation between director and multipolar theories in continuum mechanics,” Zeitschrift für angewandte Mathematik und Physik ZAMP, vol. 18, no. 2, pp. 208–218, 1967.
  • A. E. Green and R. S. Rivlin, “Multipolar continuum mechanics,” Arch. Rati. Mech. Anal., vol. 17, no. 2, pp. 113–147, 1964.
  • L. C. Martins, R. F. Oliveira, and P. Podio-Guidugli, “On the vanishing of the additive measures of strain and rotation for finite deformations,” J. Elast., vol. 17, no. 2, pp. 189–193, 1987.
  • L. C. Martins and P. Podio-Guidugli, “On the local measures of mean rotation in continuum mechanics,” J. Elast., vol. 27, no. 3, pp. 267–279, 1992.
  • W. Nowacki, Theory of Micropolar Elasticity, York, NY, USA: Springer-Verlag, 1970.
  • F. Yang, A. C. M. Chong, D. C. C. Lam, and P. Tong, “Couple stress based strain gradient theory for elasticity,” Int. J. Solids Struct., vol. 39, no. 10, pp. 2731–2743, 2002.
  • K. S. Surana, A. D. Joy, and J. N. Reddy, “A non-classical internal polar continuum theory for finite deformation of solids using first Piola-Kirchhoff stress tensor,” J. Pure Appl. Math. Adv. Appli., vol. 16, no. 1, pp. 1–41, 2016.
  • K. S. Surana, Y. Ma, A. Romkes, and J. N. Reddy, “The rate constitutive equations and their validity for progressively increasing deformation,” Mech. Adv. Mater. Struct., vol. 17, no. 7, pp. 509–533, 2010.
  • K. S. Surana, Advanced Mechanics of Continua, Boca Raton, FL, USA: CRC Press, 2015.
  • A. C. Eringen, Mechanics of Continua, Huntington, NY, USA: Robert E. Krieger Publishing Co., 1980, p. 606.
  • J. N. Reddy, An Introduction to Continuum Mechanics, Cambridge, U.K.: Cambridge University Press, 2013.
  • P. Steinmann, “A micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity,” Int. J. Solids Struct., vol. 31, no. 8, pp. 1063–1084, 1994.
  • R. D. Mindlin and H. F. Tiersten, “Effects of Couple-stresses in linear elasticity,” Arch. Ratio. Mech. Anal., vol. 11, no. 1, pp. 415–448, 1962.
  • R. A. Toupin, “Elastic materials with couple-stresses,” Arch. Ratio. Mech. Anal., vol. 11, no. 1, pp. 385–414, 1962.
  • R. D. Mindlin, “Microstructure in linear elasticity,” Arch. Ration. Mech. Anal., vol. 16, pp. 51–78.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.