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Analysis of anisotropic gradient elastic shear deformable plates

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Abstract

In this paper, Reddy’s third-order shear deformable plate theory is employed for the analysis of centrosymmetric anisotropic plate structures within strain gradient elasticity. The general three-dimensional anisotropic gradient theory is reduced to a two-dimensional formulation for the analysis of thick plate structures. The third-order shear deformation theory (TSDT) takes into account quadratic variation of the transverse shear strains of the plate and does not require shear correction factors. In order to investigate the case of small strains but moderate rotations, the von Kármán strains are considered. The TSDT is also simplified to anisotropic Kirchhoff plate theory within gradient elasticity. To study specific material properties in more detail, the (Kirchhoff and TSDT) gradient plate theory of general anisotropy is simplified to the more practical case of orthotropic plates. It is observed that the gradient theory provides the capability to capture the size effects in anisotropic plate structures. As case studies, the bending and buckling behaviors of the simply supported orthotropic (Kirchhoff and TSDT) plates are studied. Variationally consistent boundary conditions are also discussed. Finally, analytical solutions are presented for the bending and buckling of simply supported orthotropic Kirchhoff plates. The effects of internal length scales on deflections and buckling loads are presented.

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References

  1. Akgöz, B., Civalek, Ö.: A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory. Acta Mech. 226, 2277–2294 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ansari, R., Gholami, R., Mohammadi, V.: Faghih Shojaei, M.: Size-dependent pull-in instability of hydrostatically and electrostatically actuated circular microplates. J. Comp Nonlinear Dyn. 8, 021015 (2008)

    Article  Google Scholar 

  3. Ashoori, M. A., Mahmoodi, M.J.: A micro-scale modeling of Kirchhoff plate based on modified strain-gradient elasticity theory. Eu. J. Mech. A/Solids 40, 50–59 (2013)

  4. Askes, H., Aifantis, E.C.: Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int. J. Solids Struct. 48, 1962–1990 (2011)

    Article  Google Scholar 

  5. Auffray, N., Le Quang, H., He, Q.C.: Matrix representations for 3D strain-gradient elasticity. J. Mech. Phys. Solids 61, 1202–1223 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Auffray, N., Dirrenberger, J., Rosi, G.: A complete description of bi-dimensional anisotropic strain-gradient elasticity. Int. J. Solids Struct. 69–70, 195–206 (2015)

    Article  Google Scholar 

  7. Challamel, N., Hache, F., Elishakoff, I., Wang, C.M.: Buckling and vibrations of microstructured rectangular plates considering phenomenological and lattice-based nonlocal continuum models. Compos. Struct. 149, 145–156 (2016)

    Article  Google Scholar 

  8. Danesh, D., Asghari, M.: Analysis of micro-rotating disks based on the strain gradient elasticity. Acta Mech. 225, 1955–1965 (2014)

    Article  MATH  Google Scholar 

  9. Fernandes, R., Mousavi, S.M., El-Borgi, S.: Free and forced vibration nonlinear analysis of a nanobeam using finite strain and velocity gradients theory. Acta Mech (in press) (2016)

  10. Gao, X.-L., Park, S.K.: Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem. Int. J. Solids Struct. 44, 7486–7499 (2007)

    Article  MATH  Google Scholar 

  11. Gitman, I., Askes, H., Kuhl, E., Aifantis, E.: Stress concentrations in fractured compact bone simulated with a special class of anisotropic gradient elasticity. Int. J. Solids Struct. 47, 1099–1107 (2010)

    Article  MATH  Google Scholar 

  12. Giunta, G., Koutsawa, Y., Belouettar, S., Hu, H.: Static, free vibration and stability analysis of three-dimensional nano-beams by atomistic refined models accounting for surface free energy effect. Int. J. Solids Struct. 50, 1460–1472 (2013)

    Article  MATH  Google Scholar 

  13. Giunta, G., Koutsawa, Y., Belouettar, S., Hu, H.: Analysis of nano-plates by atomistic-refined models accounting for surface free energy effect. Acta Mech. 225, 31–51 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Khakalo, S., Balobanov, V., Niiranen, J.: Isogeometric Static Analysis of Gradient-Elastic Plane Strain/Stress Problems. Generalized Continua as Models for Classical and Advanced Materials. 42, 229–235 (2016)

    Article  Google Scholar 

  15. Lazar, M., Maugin, G.A.: Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity. Int. J. Eng. Sci. 43, 1157–1184 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lazar, M., Po, G.: The non-singular Green tensor of Mindlin’s anisotropic gradient elasticity with separable weak non-locality. Phys. Lett. A 379, 1538–1543 (2015)

  17. Lazar, M., Po, G.: The non-singular Green tensor of gradient anisotropic elasticity of Helmholtz type. Eur. J. Mech. A/Solids 50, 152–162 (2015)

  18. Lazopoulos, K.A.: On bending of strain gradient elastic micro-plates. Mech. Res. Commun. 36, 777–783 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, A., Zhou, S., Zhou, S., Wang, B.: A size-dependent model for bi-layered Kirchhoff micro-plate based on strain gradient elasticity theory. Compos. Struct. 113, 272–280 (2014)

    Article  Google Scholar 

  20. Maugin, G.A.: Continuum Mechanics Through the Twentieth Century: A Concise Historical Perspective. Springer, New York (2013)

    Book  MATH  Google Scholar 

  21. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mousavi, S.M., Paavola, J., Reddy, J.N.: Variational approach to dynamic analysis of third-order shear deformable plates within gradient elasticity. Meccanica 50, 1537–1550 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mousavi, S.M., Paavola, J.: Analysis of plate in second gradient elasticity. Arch. Appl. Mech. 84, 1135–1143 (2014)

    Article  MATH  Google Scholar 

  24. Nazemnezhad, R., Zare, M.: Nonlocal Reddy beam model for free vibration analysis of multilayer nanoribbons incorporating interlayer shear effect. Eur. J. Mech. A/Solids 55, 234–242 (2016)

    Article  MathSciNet  Google Scholar 

  25. Nye, J.F.: Physical Properties of Crystals. Oxford University Press, Oxford (1957)

    MATH  Google Scholar 

  26. Ojaghnezhad, F., Shodja, H.M.: Surface elasticity revisited in the context of second strain gradient theory. Mech. Mater. 93, 220–237 (2016)

    Article  Google Scholar 

  27. Papargyri-Beskou, S., Giannakopoulos, A.E., Beskos, D.E.: Variational analysis of gradient elastic flexural plates under static loading. Int. J. Solids Struct. 47, 2755–2766 (2010)

    Article  MATH  Google Scholar 

  28. Ramezani, S.: A shear deformation micro-plate model based on the most general form of strain gradient elasticity. Int. J. Mech. Sci. 57, 34–42 (2012)

    Article  Google Scholar 

  29. Reddy, J.N.: A simple higher-order theory for laminated composite plates. J. Appl. Mech. 51, 745–752 (1984)

    Article  MATH  Google Scholar 

  30. Reddy, J.N.: Energy Principles and Variational Methods in Applied Mechanics, 2nd edn. Wiley, New York (2002)

    Google Scholar 

  31. Reddy, J.N.: Theory and Analysis of Elastic Plates and Shells, 2nd edn. CRC Press, Boca Raton (2007)

    Google Scholar 

  32. Reddy, J.N.: An Introduction to Continuum Mechanics, 2nd edn. Cambridge University Press, New York (2013)

    Google Scholar 

  33. Reddy, J.N.: An Introduction to Nonlinear Finite Element Analysis, 2nd edn. Oxford University Press, Oxford (2015)

    Google Scholar 

  34. Seif, D., Po, G., Mrovec, M., Lazar, M., Elsässer, C., Gumbsch, P.: Atomistically enabled nonsingular anisotropic elastic representation of near-core dislocation stress fields in \(\alpha \)-iron. Phys. Rev. B 91, 184102 (2015)

    Article  Google Scholar 

  35. Tahaei Yaghoubi, S., Mousavi, S.M., Paavola, J.: Strain and velocity gradient theory for higher-order shear deformable beams. Arch. Appl. Mech. 85, 877–892 (2015)

  36. Tahaei Yaghoubi, S., Mousavi S.M., Paavola J.: Buckling of centrosymmetric anisotropic beam structures within strain gradient elasticity (2016) (under review)

  37. Voigt, W.: Lehrbuch der Kristallphysik, reprint of the 1st edn. Teubner, Leipzig (1928)

  38. Xu, K.J., Deng, Z.C., Meng, J.M., Zhang, K.: Bending and vibration analysis of generalized gradient elastic plates. Acta Mech. 225, 3463–3482 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Civil Engineering, Aalto University, Box 12100, 00076, Aalto, Finland

    S. M. Mousavi

  2. Department of Mechanical Engineering, Texas A&M University, College Station, TX, 77843-3123, USA

    J. N. Reddy

  3. Department of Mechanical Engineering, Aalto University, Box 15300, 00076, Aalto, Finland

    J. Romanoff

Authors
  1. S. M. Mousavi
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  2. J. N. Reddy
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  3. J. Romanoff
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Correspondence to S. M. Mousavi.

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Mousavi, S.M., Reddy, J.N. & Romanoff, J. Analysis of anisotropic gradient elastic shear deformable plates. Acta Mech 227, 3639–3656 (2016). https://doi.org/10.1007/s00707-016-1689-z

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  • DOI: https://doi.org/10.1007/s00707-016-1689-z

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