ORIGINAL PAPER
Size-dependent Kirchhoff plates based on deviatoric couple stress elasticity
Ya-Wei Wang,
Jian Chen,
Xian-Fang Li,
Ya-Wei Wang
School of Civil Engineering, Central South University, Changsha, PR China
Search for more papers by this authorJian Chen
State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing, PR China
Search for more papers by this authorCorresponding Author
Xian-Fang Li
School of Civil Engineering, Central South University, Changsha, PR China
Correspondence
Xian-Fang Li, School of Civil Engineering, Central South University, Changsha 410075, PR China.
Email: [email protected]
Search for more papers by this authorYa-Wei Wang,
Jian Chen,
Xian-Fang Li,
Ya-Wei Wang
School of Civil Engineering, Central South University, Changsha, PR China
Search for more papers by this authorJian Chen
State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing, PR China
Search for more papers by this authorCorresponding Author
Xian-Fang Li
School of Civil Engineering, Central South University, Changsha, PR China
Correspondence
Xian-Fang Li, School of Civil Engineering, Central South University, Changsha 410075, PR China.
Email: [email protected]
Search for more papers by this authorFirst published: 21 May 2025
Abstract
The deviatoric couple stress theory (DCST) is introduced to analyze the statics and dynamics behaviors of thin microplates. A size-dependent Kirchhoff plate theory is formulated, and the governing differential equation and appropriate boundary conditions are established. The static bending, including pure bending, of rectangular microplates, and the free vibration of circular microplates are studied and the size-dependence of the mechanical behaviors of thin microplates is demonstrated utilizing the DCST. The obtained results indicate that the natural frequency and maximum deflection are significantly influenced by the scale parameters of the DCST.
REFERENCES
- 1Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory and experiment. Acta Mater. 42(2), 475–487 (1994)
- 2Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J.-X., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51(8), 1477–1508 (2003)
- 3Haque, M.A., Saif, M.T.A.: Strain gradient effect in nanoscale thin films. Acta Mater. 51(11), 3053–3061 (2003)
- 4Shorkin, V.S., Vilchevskaya, E.N., Altenbach, H.: Linear theory of micropolar media with internal nonlocal potential interactions. ZAMM - J. Appl. Math. Mech. 103(11), e202300099 (2023)
10.1002/zamm.202300099 Google Scholar
- 5Eghbali, M., Hosseini, S.A., Pourseifi, M.: Free transverse vibrations analysis of size-dependent cracked piezoelectric nano-beam based on the strain gradient theory under mechanic-electro forces. Eng. Anal. Bound. Elem. 143(03), 606–612 (2022)
10.1016/j.enganabound.2022.07.006 Google Scholar
- 6Eghbali, M., Hosseini, S.A., Pourseifi, M.: A dynamical evaluation of size-dependent weakened nano-beam based on the nonlocal strain gradient theory. J. Strain Anal. Eng. Des. 58(5), 357–366 (2023)
10.1177/03093247221135210 Google Scholar
- 7Aouadi, M., Passarella, F., Tibullo, V.: Stabilization in extensible thermoelastic Timoshenko microbeam based on modified couple stress theory. J. Appl. Math. Mech. 104(4), e202300787 (2024)
10.1002/zamm.202300787 Google Scholar
- 8Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11(1), 415–448 (1962)
10.1007/BF00253946 Google Scholar
- 9Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39(10), 2731–2743 (2002)
- 10Hadjesfandiari, A.R., Dargush, G.F.: Couple stress theory for solids. Int. J. Solids Struct. 48(18), 2496–2510 (2011)
- 11Gupta, A., Jain, N.K., Salhotra, R., Joshi, P.V.: Effect of microstructure on vibration characteristics of partially cracked rectangular plates based on a modified couple stress theory. Int. J. Mech. Sci. 100, 269–282 (2015)
- 12Chen, J., Wang, Y.-W., Li, X.-F.: A mode -I crack embedded in a prestressed material with microstructure. Eur. J. Mech. A. Solids 100, 104990 (2023)
- 13Chen, J., Wang, Y.-W., Zheng, R.-Y., Li, X.-F.: Mode-III interface crack in a bi-material with initial stress and couple stress. Eng. Fract. Mech. 281, 109135 (2023)
- 14Wang, Y.X., Zhang, X., Shen, H.M., Liu, J., Zhang, B.: Couple stress -based 3D contact of elastic films. Int. J. Solids Struct. 191–192, 449–463 (2020)
10.1016/j.ijsolstr.2020.01.005 Google Scholar
- 15Radi, E.: A loaded beam in full frictionless contact with a couple stress elastic half-plane: Effects of non-standard contact conditions. Int. J. Solids Struct. 232, 111175 (2021)
- 16Eyvazian, A., Zhang, C.W., Alkhedher, M., Muhsen, S., Elkotb, M.A.: Thermal buckling and post-buckling analyses of rotating Timoshenko microbeams reinforced with graphene platelet. Compos. Struct. 304, 116358 (2023)
- 17Wang, C., Chen, X., Wei, P., Li, Y.: Reflection and transmission of elastic waves through a couple-stress elastic slab sandwiched between two half-spaces. Acta Mech. Solid Sin. 33(6), 1022–1039 (2017)
- 18Liu, C.C., Yu, J.G., Xu, W.J., Zhang, X.M., Wang, X.H.: Dispersion characteristics of guided waves in functionally graded anisotropic micro/nano-plates based on the modified couple stress theory. Thin-Wall. Struct. 161, 107527 (2021)
- 19Wang, Y.-W., Li, X.-F.: Synergistic effect of memory-size-microstructure on thermoelastic damping of a micro-plate. Int. J. Heat Mass Transfer 181, 122031 (2021)
- 20Park, S.K., Gao, X.-L.: Bernoulli-Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16(11), 2355 (2006)
- 21Chen, W.J., Li, L., Xu, M.: A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation. Compos. Struct. 93(11), 2723–2732 (2011)
- 22Ma, 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(12), 3379–3391 (2008)
- 23Tsiatas, G.C.: A new Kirchhoff plate model based on a modified couple stress theory. Int. J. Solids Struct. 46(13), 2757–2764 (2009)
- 24Ma, H.M., Gao, X.-L., Reddy, J.N.: A non -classical Mindlin plate model based on a modified couple stress theory. Acta Mech. 220(1–4), 217–235 (2011)
- 25Wang, Y.-W., Chen, J., Li, X.-F.: Deviatoric couple stress theory and its application to simple shear and pure bending problems. Appl. Math. Modell. 138, 115799 (2025)
- 26Gao, X.-L., Zhang, G.Y.: A non-classical Kirchhoff plate model incorporating microstructure, surface energy and foundation effects. Contin. Mech. Thermodyn. 28, 195–213 (2016)
- 27Zhou, Y., Huang, K.: A simplified deformation gradient theory and its experimental verification. Acta Mech. 234(7), 2963–2984 (2023)
- 28Shaat, M., Mahmoud, F.F., Gao, X.L., Faheem, A.F.: Size-dependent bending analysis of Kirchhoff nano-plates based on a modified couple-stress theory including surface effects. Int. J. Mech. Sci. 79, 31–37 (2014)
- 29Thai, H.-T., Choi, D.-H.: Size-dependent functionally graded Kirchhoff and Mindlin plate models based on a modified couple stress theory. Compos. Struct. 95, 142–153 (2013)
- 30Sladek, V., Sladek, J., Repka, M., Sator, L.: FGM micro/nano-plates within modified couple stress elasticity. Compos. Struct. 245, 112294 (2020)
- 31Yang, Y., Li, X.-F.: Bending and free vibration of a circular magnetoelectroelastic plate with surface effects. Int. J. Mech. Sci. 157–158, 858–871 (2019)
10.1016/j.ijmecsci.2019.05.029 Google Scholar
- 32Wang, B.L., Zhou, S.J., Zhao, J.F., Chen, X.: A size -dependent Kirchhoff micro-plate model based on strain gradient elasticity theory. Eur. J. Mech. A. Solids 30(4), 517–524 (2011)
- 33Movassagh, A.A., Mahmoodi, M.J.: A micro-scale modeling of Kirchhoff plate based on modified strain-gradient elasticity theory. Eur. J. Mech. A. Solids 40, 50–59 (2013)
- 34Wu, C.-P., Hu, H.-X.: A unified size-dependent plate theory for static bending and free vibration analyses of micro-and nano-scale plates based on the consistent couple stress theory. Mech. Mater. 162, 104085 (2021)
- 35Koiter, W.T.: Couple stresses in the theory of elasticity, i and ii. Proc. Ned. Akad. Wet. (B) 67(1), 17–29 (1964)
- 36Li, Z.K., He, Y.M., Lei, J., Guo, S., Liu, D.B., Wang, L.: A standard experimental method for determining the material length scale based on modified couple stress theory. Int. J. Mech. Sci. 141, 198–205 (2018)
- 37Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics, 3rd ed., Springer, Berlin (2004).
10.1007/978-3-662-10388-3 Google Scholar
- 38Reddy, J.N.: Theory and Analysis of Elastic Plates and Shells, CRC Press, Boca Raton (2006)
10.1201/9780849384165 Google Scholar
- 39Timoshenko, S.P., Woinowsky-Krieger, S.: Theory of Plates and Shells, McGraw-hill, New York (1959)
- 40Kim, J., Zur, K.K., Reddy, J.N.: Bending, free vibration, and buckling of modified couples stress-based functionally graded porous micro-plates. Compos. Struct. 209, 879–888 (2019)
- 41Wang, Y.-W., Chen, J., Zheng, R.-Y., Li, X.-F.: Thermoelastic damping in circular microplate resonators based on fractional dual-phase-lag model and couple stress theory. Int. J. Heat Mass Transfer 201, 123570 (2023)
- 42Reddy, J.N.: Energy and Variational Methods in Applied Mechanics, Wiley, Hoboken, NJ (2017)
- 43Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975)
- 44Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147 (2000)
- 45McFarland, A.W., Colton, J.S.: Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J. Micromech. Microeng. 15(5), 1060 (2005)
- 46Leissa, A.W.: Vibration of Plates, National Aeronautics and Space Administration, Washington, DC (1969)
- 47Lu, L., Guo, X.M., Zhao, J.Z.: A unified size-dependent plate model based on nonlocal strain gradient theory including surface effects. Appl. Math. Modell. 68, 583–602 (2019)
- 48Hosseini-Hashemi, S., Zare, M., Nazemnezhad, R.: An exact analytical approach for free vibration of Mindlin rectangular nano-plates via nonlocal elasticity. Compos. Struct. 100, 290–299 (2013)
- 49Yang, Y., Hu, Z.-L., Li, X.-F.: Axisymmetric bending and vibration of circular nanoplates with surface stresses. Thin-Wall. Struct. 166, 108086 (2021)
