Abstract
A non-classical model for an orthotropic Kirchhoff plate embedded in a viscoelastic medium is developed by using an extended version of the modified couple stress theory and a three-parameter foundation model. The equations of motion and the boundary conditions are simultaneously obtained through a variational formulation based on Hamilton’s principle. The new plate model contains three material length scale parameters to capture the microstructure effect, one damping coefficient to account for the viscoelastic damping effect, and two foundation moduli to represent the foundation effect. The current non-classical orthotropic plate model includes the isotropic plate model incorporating the microstructure effect and the classical elasticity-based orthotropic plate model as special cases. To illustrate the new model, the static bending and free vibration problems of a simply supported orthotropic plate are analytically solved by directly applying the general formulas derived. For the static bending problem, the numerical results reveal that the deflection of the simply supported plate predicted by the current model is smaller than that predicted by the classical model, and the difference is large when the plate thickness is small but diminishes as the thickness becomes large. For the free vibration problem, it is found that the natural frequency predicted by the new plate model with or without the foundation is higher than that predicted by the classical plate model, and the difference is significant for very thin plates. In addition, the damping ratio predicted by the new plate model is lower than that predicted by the classical plate model, and the difference is diminishing as the plate thickness increases. These predicted trends of the size effect at the micron scale agree with those observed experimentally. Furthermore, it is quantitatively shown that the presence of the foundation enlarges the plate natural frequency.
Similar content being viewed by others
References
Jones, R.M.: Mechanics of Composite Materials. Taylor & Francis, New York (1999)
Arani, A.G., Jalaei, M.H.: Transient behavior of an orthotropic graphene sheet resting on orthotropic visco-Pasternak foundation. Int. J. Eng. Sci. 103, 97–113 (2016)
Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147 (2000)
Lim, C.W., He, L.H.: Size-dependent nonlinear response of thin elastic films with nano-scale thickness. Int. J. Mech. Sci. 46, 1715–1726 (2004)
McFarland, A.W., Colton, J.S.: Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J. Micromech. Microeng. 15, 1060–1067 (2005)
Lazopoulos, K.A.: On the gradient strain elasticity theory of plates. Eur. J. Mech. A/Solids 23, 843–852 (2004)
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)
Gao, X.-L., Ma, H.M.: Solution of Eshelby’s inclusion problem with a bounded domain and Eshelby’s tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory. J. Mech. Phys. Solids 58, 779–797 (2010)
Gao, X.-L., Zhou, S.-S.: Strain gradient solutions of half-space and half-plane contact problems. Z. Angew. Math. Phys. 64, 1363–1386 (2013)
Lazar, M., Maugin, G.A., Aifantis, E.C.: On dislocations in a special class of generalized elasticity. Phys. Status Solidi (b) 242, 2365–2390 (2005)
Gourgiotis, P.A., Georgiadis, H.G.: Plane-strain crack problems in microstructured solids governed by dipolar gradient elasticity. J. Mech. Phys. Solids 57, 1898–1920 (2009)
Papargyri-Beskou, S., Beskos, D.E.: Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates. Arch. Appl. Mech. 78, 625–635 (2008)
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)
Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)
Lu, P., Zhang, P.Q., Lee, H.P., Wang, C.M., Reddy, J.N.: Non-local elastic plate theories. Proc. R. Soc. A 463, 3225–3240 (2007)
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)
Park, S.K., Gao, X.-L.: Variational formulation of a modified couple stress theory and its application to a simple shear problem. Z. Angew. Math. Phys. 59, 904–917 (2008)
Tsiatas, G.C.: A new Kirchhoff plate model based on a modified couple stress theory. Int. J. Solids Struct. 46, 2757–2764 (2009)
Jomehzadeh, E., Noori, H.R., Saidi, A.R.: The size-dependent vibration analysis of micro-plates based on a modified couple stress theory. Phys. E 43, 877–883 (2011)
Akgöz, B., Civalek, Ö.: Modeling and analysis of micro-sized plates resting on elastic medium using the modified couple stress theory. Meccanica 48, 863–873 (2013)
Ma, H.M., Gao, X.-L., Reddy, J.N.: A non-classical Mindlin plate model based on a modified couple stress theory. Acta Mech. 220, 217–235 (2011)
Gao, X.-L., Huang, J.X., Reddy, J.N.: A non-classical third-order shear deformation plate model based on a modified couple stress theory. Acta Mech. 224, 2699–2718 (2013)
Zhou, S.-S., Gao, X.-L.: A non-classical model for circular Mindlin plates based on a modified couple stress theory. ASME J. Appl. Mech. 81, 051014-1–051014-8 (2014)
Zhang, G.Y., Gao, X.-L., Wang, J.Z.: A non-classical model for circular Kirchhoff plates incorporating microstructure and surface energy effects. Acta Mech. 226, 4073–4085 (2015)
Gao, X.-L., Zhang, G.Y.: A non-classical Kirchhoff plate model incorporating microstructure, surface energy and foundation effects. Continuum Mech. Thermodyn. 28, 195–213 (2016)
Gao, X.-L., Zhang, G.Y.: A non-classical Mindlin plate model incorporating microstructure, surface energy and foundation effects. Proc. R. Soc. A 472, 20160275-1–20160275-25 (2016)
Zhang, G.Y., Gao, X.-L., Tang, S.: A non-classical model for circular Mindlin plates incorporating microstructure and surface energy effects. Procedia IUTAM 21, 48–55 (2017)
Pradhan, S.C., Kumar, A.: Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method. Compos. Struct. 93, 774–779 (2011)
Hosseini, M., Bahreman, M., Jamalpoor, A.: Using the modified strain gradient theory to investigate the size-dependent biaxial buckling analysis of an orthotropic multi-microplate system. Acta Mech. 227, 1621–1643 (2016)
Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)
Tsiatas, G.C., Yiotis, A.J.: Size effect on the static, dynamic and buckling analysis of orthotropic Kirchhoff-type skew micro-plates based on a modified couple stress theory: comparison with the nonlocal elasticity theory. Acta Mech. 226, 1267–1281 (2015)
Guo, J., Chen, J., Pan, E.: Analytical three-dimensional solutions of anisotropic multilayered composite plates with modified couple-stress effect. Compos. Struct. 153, 321–331 (2016)
Guo, J., Chen, J., Pan, E.: Size-dependent behavior of functionally graded anisotropic composite plates. Int. J. Eng. Sci. 106, 110–124 (2016)
Chen, W., Li, X.: A new modified couple stress theory for anisotropic elasticity and microscale laminated Kirchhoff plate model. Arch. Appl. Mech. 84, 323–341 (2014)
Mindlin, R.D.: Influence of couple-stresses on stress concentrations. Exp. Mech. 3, 1–7 (1963)
Koiter, W.T.: Couple-stresses in the theory of elasticity. Proc. K. Ned. Akad. Wet. B 67, 17–44 (1964)
Lakes, R.S., Benedict, R.L.: Noncentrosymmetry in micropolar elasticity. Int. J. Eng. Sci. 20, 1161–1167 (1982)
Lazar, M., Kirchner, H.O.K.: Cosserat (micropolar) elasticity in Stroh form. Int. J. Solids Struct. 42, 5377–5398 (2005)
Iesan, D., Scalia, A.: On the deformation of orthotropic Cosserat elastic cylinders. Math. Mech. Solids 16, 177–199 (2010)
Goda, I., Assidi, M., Ganghoffer, J.F.: A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure. Biomech. Model. Mechanobiol. 13, 53–83 (2014)
Ghiba, I.D., Neff, P., Madeo, A., Münch, I.: A variant of the linear isotropic indeterminate couple-stress model with symmetric local force-stress, symmetric nonlocal force-stress, symmetric couple-stresses and orthogonal boundary conditions. Math. Mech. Solids. 22, 1221–1266 (2017)
Reddy, J.N.: Energy Principles and Variational Methods in Applied Mechanics, 2nd edn. Wiley, Hoboken, NJ (2002)
Selvadurai, A.P.S.: Elastic Analysis of Soil-Foundation Interaction. Elsevier, Amsterdam (1979)
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, 3379–3391 (2008)
Ma, H.M., Gao, X.-L., Reddy, J.N.: A non-classical Reddy–Levinson beam model based on a modified couple stress theory. Int. J. Multiscale Comput. Eng. 8, 167–180 (2010)
Gao, X.-L.: A new Timoshenko beam model incorporating microstructure and surface energy effects. Acta Mech. 226, 457–474 (2015)
Gao, X.-L., Zhang, G.Y.: A microstructure- and surface energy-dependent third-order shear deformation beam model. Z. Angew. Math. Phys. 66, 1871–1894 (2015)
Gao, X.-L., Mall, S.: Variational solution for a cracked mosaic model of woven fabric composites. Int. J. Solids Struct. 38, 855–874 (2001)
Xing, Y.F., Liu, B.: New exact solutions for free vibrations of thin orthotropic rectangular plates. Compos. Struct. 89, 567–574 (2009)
Joshi, P.V., Jain, N.K., Ramtekkar, G.D.: Analytical modelling for vibration analysis of partially cracked orthotropic rectangular plates. Euro. J. Mech. A/Solids 50, 100–111 (2015)
Park, S.K., Gao, X.-L.: Bernoulli–Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16, 2355–2359 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, G.Y., Gao, XL. & Guo, Z.Y. A non-classical model for an orthotropic Kirchhoff plate embedded in a viscoelastic medium. Acta Mech 228, 3811–3825 (2017). https://doi.org/10.1007/s00707-017-1906-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-017-1906-4