Variational formulation and differential quadrature finite element for freely vibrating strain gradient Kirchhoff plates. (English) Zbl 07813064
Summary: In this paper, we apply the energy variational principle to arrive at the differential equation of motion and all appropriate boundary conditions for strain gradient Kirchhoff micro-plates. The resulting sixth-order boundary value problem of free vibration is solved by a thirty-six-DOF four-node differential quadrature plate finite element. The \(C^2\)-continuity condition of the deflection is guaranteed by devising a sixth-order differential quadrature-based geometric mapping scheme that can transform the displacement parameters at Gauss-Lobatto quadrature points into those at element nodes. The total potential energy of a generic micro-plate element is firstly discretized in terms of nodal parameters. It is then minimized to obtain the formulation of element stiffness and mass matrices. For comparison reasons, a Hermite interpolation-based strain gradient finite element is provided. With the help of the symbolic computation system Maple, the explicit algebraic relationship between the stiffness (or mass) matrices of two types of elements is derived. Convergence and comparison studies are conducted to show the efficacy of our element in the free vibration analysis of macro/micro- plates. Finally, we apply the developed method to study the size-dependent vibration behavior of micro-plates with uniform or stepped thickness. Numerical examples reveal that strain gradient effects can change the vibration mode shapes, not the vibration frequencies alone.
© 2020 Wiley-VCH GmbH
© 2020 Wiley-VCH GmbH
MSC:
| 74Kxx | Thin bodies, structures |
| 74Sxx | Numerical and other methods in solid mechanics |
| 74Axx | Generalities, axiomatics, foundations of continuum mechanics of solids |
Keywords:
\(C^2\)-continuity; differential quadrature plate finite element; energy variational principle; strain gradient Kirchhoff micro-plates; vibrationReferences:
| [1] | Choudhary, V., Iniewski, K.: Mems: Fundamental Technology and Applications, CRC Press (2016) |
| [2] | Lam, D.C., Yang, F., Chong, A., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids51(8), 1477-1508 (2003) · Zbl 1077.74517 |
| [3] | Lei, J., He, Y., Guo, S., Li, Z., Liu, D.: Size‐dependent vibration of nickel cantilever microbeams: Experiment and gradient elasticity. AIP Adv.6(10), 105202 (2016) |
| [4] | Li, Z., He, Y., Lei, J., Guo, S., Liu, D., 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) |
| [5] | Li, Z., He, Y., Zhang, B., Lei, J., Guo, S., Liu, D.: Experimental investigation and theoretical modelling on nonlinear dynamics of cantilevered microbeams. Eur. J. Mech. A. Solids103834 (2019) · Zbl 1483.74042 |
| [6] | Liu, D., He, Y., Tang, X., Ding, H., Hu, P., Cao, P.: Size effects in the torsion of microscale copper wires: Experiment and analysis. Scr. Mater.66(6), 406-409 (2012) |
| [7] | Liu, D., He, Y., Dunstan, D., Zhang, B., Gan, Z., Hu, P., Ding, H.: Anomalous plasticity in the cyclic torsion of micron scale metallic wires. Phys. Rev. Lett.110(24), 244301 (2013) |
| [8] | Polyzos, D., Huber, G., Mylonakis, G., Triantafyllidis, T., Papargyri‐Beskou, S., Beskos, D.: Torsional vibrations of a column of fine‐grained material: a gradient elastic approach. J. Mech. Phys. Solids76, 338-358 (2015) · Zbl 1349.74183 |
| [9] | Guo, S., He, Y., Tian, M., Liu, D., Li, Z., Lei, J., Han, S.: Size effect in cyclic torsion of micron‐scale polycrystalline copper wires. Mater. Sci. Eng., A792, 139671 (2020) |
| [10] | Vladimir, E.: Wave Processes in Solids with Microstructure, World Scientific (2003) · Zbl 1056.74001 |
| [11] | Mindlin, R., Tiersten, H.: Effects of couple‐stresses in linear elasticity. Arch. Ration. Mech. Anal.11(1), 415-448 (1962) · Zbl 0112.38906 |
| [12] | Toupin, R.: Elastic materials with couple‐stresses. Arch. Ration. Mech. Anal.11(1), 385-414 (1962) · Zbl 0112.16805 |
| [13] | Yang, F., Chong, A., Lam, D.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct.39(10), 2731-2743 (2002) · Zbl 1037.74006 |
| [14] | Hadjesfandiari, A.R., Dargush, G.F.: Couple stress theory for solids. Int. J. Solids Struct.48(18), 2496-2510 (2011) |
| [15] | Mindlin, R.: Micro‐structure in linear elasticity. Arch. Ration. Mech. Anal.16(1), 51-78 (1964) · Zbl 0119.40302 |
| [16] | Mindlin, R.D.: Second gradient of strain and surface‐tension in linear elasticity. Int. J. Solids Struct.1(4), 417-438 (1965) |
| [17] | Mindlin, R., Eshel, N.: On first strain‐gradient theories in linear elasticity. Int. J. Solids Struct.4(1), 109-124 (1968) · Zbl 0166.20601 |
| [18] | Altan, B.S., Aifantis, E.C.: On some aspects in the special theory of gradient elasticity. J. Mech. Behav. Mater.8(3) (1997) |
| [19] | Polizzotto, C.: A gradient elasticity theory for second‐grade materials and higher order inertia. Int. J. Solids Struct.49(15‐16), 2121-2137 (2012) |
| [20] | Cordero, N.M., Forest, S., Busso, E.P.: Second strain gradient elasticity of nano‐objects. J. Mech. Phys. Solids97, 92-124 (2016) |
| [21] | Münch, I., Neff, P., Madeo, A., Ghiba, I.D.: The modified indeterminate couple stress model: Why Yang et al.’s arguments motivating a symmetric couple stress tensor contain a gap and why the couple stress tensor may be chosen symmetric nevertheless. J. Appl. Math. Mech.97(12), 1524-1554 (2017) · Zbl 1538.74002 |
| [22] | Lazar, M.: Irreducible decomposition of strain gradient tensor in isotropic strain gradient elasticity. J. Appl. Math. Mech.96(11), 1291-1305 (2016) · Zbl 07775124 |
| [23] | Lazar, M., Maugin, G.A.: Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity. Int. J. Eng. Sci.43(13/14), 1157-1184 (2005) · Zbl 1211.74040 |
| [24] | Froiio, F., Zervos, A.: Second‐grade elasticity revisited. Math. Mech. Solids24(3), 748-777 (2019) · Zbl 1444.74008 |
| [25] | Ma, H., Gao, X.‐L., Reddy, J.: A microstructure‐dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids56(12), 3379-3391 (2008) · Zbl 1171.74367 |
| [26] | Kong, S., Zhou, S., Nie, Z., Wang, K.: Static and dynamic analysis of micro beams based on strain gradient elasticity theory. Int. J. Eng. Sci.47(4), 487-498 (2009) · Zbl 1213.74190 |
| [27] | Wang, B., Zhao, J., Zhou, S.: A micro scale Timoshenko beam model based on strain gradient elasticity theory. Eur. J. Mech. A. Solids29(4), 591-599 (2010) · Zbl 1480.74194 |
| [28] | Lazopoulos, K., Lazopoulos, A.: On a strain gradient elastic Timoshenko beam model. J. Appl. Math. Mech.91(11), 875-882 (2011) · Zbl 1280.74020 |
| [29] | Şimşek, M., Reddy, J.: Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory. Int. J. Eng. Sci.64, 37-53 (2013) · Zbl 1423.74517 |
| [30] | Zhang, B., He, Y., Liu, D., Gan, Z., Shen, L.: A novel size‐dependent functionally graded curved mircobeam model based on the strain gradient elasticity theory. Compos. Struct.106, 374-392 (2013) |
| [31] | Zhang, B., He, Y., Liu, D., Gan, Z., Shen, L.: Size‐dependent functionally graded beam model based on an improved third‐order shear deformation theory. Eur. J. Mech. A. Solids47, 211-230 (2014) · Zbl 1406.74416 |
| [32] | Aghazadeh, R., Cigeroglu, E., Dag, S.: Static and free vibration analyses of small‐scale functionally graded beams possessing a variable length scale parameter using different beam theories. Eur. J. Mech. A. Solids46 (2014) · Zbl 1406.74366 |
| [33] | Li, A., Zhou, S., Zhou, S., Wang, B.: A size‐dependent bilayered microbeam model based on strain gradient elasticity theory. Compos. Struct.108, 259-266 (2014) |
| [34] | Khodabakhshi, P., Reddy, J.: A unified beam theory with strain gradient effect and the von Kármán nonlinearity. J. Appl. Math. Mech.97(1), 70-91 (2017) · Zbl 07775115 |
| [35] | Tahaei Yaghoubi, S., Balobanov, V., Mousavi, S.M., Niiranen, J.: Variational formulations and isogeometric analysis for the dynamics of anisotropic gradient‐elastic Euler‐Bernoulli and shear‐deformable beams. Eur. J. Mech. A. Solids69, 113-123 (2018) · Zbl 1406.74409 |
| [36] | Zhang, B., Shen, H., Liu, J., Wang, Y., Zhang, Y.: Deep postbuckling and nonlinear bending behaviors of nanobeams with nonlocal and strain gradient effects. Appl. Math. Mech.1-34 (2019) |
| [37] | Jin, M., Qi, H.: Initial post‐buckling of the micro‐scaled beams based on the strain gradient elasticity. J. Appl. Math. Mech. (2019) · Zbl 07804550 |
| [38] | Amir, S., Soleimani‐Javid, Z., Arshid, E.: Size‐dependent free vibration of sandwich micro beam with porous core subjected to thermal load based on SSDBT. J. Appl. Math. Mech. (2019) · Zbl 07804541 |
| [39] | Fu, G., Zhou, S., Qi, L.: A size‐dependent Bernoulli‐Euler beam model based on strain gradient elasticity theory incorporating surface effects. J. Appl. Math. Mech.99(6), e201800048 (2019) · Zbl 1538.74082 |
| [40] | Papargyri‐Beskou, S., Giannakopoulos, A., Beskos, D.: Variational analysis of gradient elastic flexural plates under static loading. Int. J. Solids Struct.47(20), 2755-2766 (2010) · Zbl 1196.74093 |
| [41] | Wang, B., Zhou, S., Zhao, J., Chen, X.: A size‐dependent Kirchhoff micro‐plate model based on strain gradient elasticity theory. Eur. J. Mech. A. Solids30(4), 517-524 (2011) · Zbl 1278.74103 |
| [42] | Reddy, J., Kim, J.: A nonlinear modified couple stress‐based third‐order theory of functionally graded plates. Compos. Struct.94(3), 1128-1143 (2012) |
| [43] | Ke, L.‐L., Wang, Y.‐S., Yang, J., Kitipornchai, S.: Free vibration of size‐dependent Mindlin microplates based on the modified couple stress theory. J. Sound Vib.331(1), 94-106 (2012) |
| [44] | Zhang, B., He, Y., Liu, D., Lei, J., Shen, L., Wang, L.: A size‐dependent third‐order shear deformable plate model incorporating strain gradient effects for mechanical analysis of functionally graded circular/annular microplates. Composites, Part B79, 553-580 (2015) |
| [45] | Lei, J., He, Y., Zhang, B., Liu, D., Shen, L., Guo, S.: A size‐dependent FG micro‐plate model incorporating higher‐order shear and normal deformation effects based on a modified couple stress theory. Int. J. Mech. Sci.104, 8-23 (2015) |
| [46] | Ansari, R., Gholami, R., Faghih Shojaei, M., Mohammadi, V., Sahmani, S.: Bending, buckling and free vibration analysis of size‐dependent functionally graded circular/annular microplates based on the modified strain gradient elasticity theory. Eur. J. Mech. A. Solids49, 251-267 (2015) · Zbl 1406.74526 |
| [47] | Zhang, B., He, Y., Liu, D., Shen, L., Lei, J.: An efficient size‐dependent plate theory for bending, buckling and free vibration analyses of functionally graded microplates resting on elastic foundation. Appl. Math. Modell.39(13), 3814-3845 (2015) · Zbl 1443.74228 |
| [48] | Eshraghi, I., Dag, S., Soltani, N.: Consideration of spatial variation of the length scale parameter in static and dynamic analyses of functionally graded annular and circular micro‐plates. Composites, Part B (2015) |
| [49] | Eshraghi, I., Dag, S., Soltani, N.: Bending and free vibrations of functionally graded annular and circular micro‐plates under thermal loading. Compos. Struct.137, 196-207 (2016) |
| [50] | Ghassabi, A.A., Dag, S., Cigeroglu, E.: Free vibration analysis of functionally graded rectangular nanoplates considering spatial variation of the nonlocal parameter. Arch. Mech.69(2), 105-130 (2017) · Zbl 1386.74062 |
| [51] | Tahaei Yaghoubi, S., Mousavi, S.M., Paavola, J.: Size effects on centrosymmetric anisotropic shear deformable beam structures. J. Appl. Math. Mech.97(5), 586-601 (2017) · Zbl 07777496 |
| [52] | Zozulya, V.: Higher order couple stress theory of plates and shells. J. Appl. Math. Mech.98(10), 1834-1863 (2018) · Zbl 07776935 |
| [53] | Aghazadeh, R., Dag, S., Cigeroglu, E.: Thermal effect on bending, buckling and free vibration of functionally graded rectangular micro‐plates possessing a variable length scale parameter. Microsyst. Technol. (2018) |
| [54] | Aghazadeh, R., Dag, S., Cigeroglu, E.: Modelling of graded rectangular micro‐plates with variable length scale parameters. Struct. Eng. Mech.65(5), 573-585 (2018) |
| [55] | dell’Isola, F., Madeo, A., Placidi, L.: Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua. J. Appl. Math. Mech.92(1), 52-71 (2012) · Zbl 1247.74031 |
| [56] | Dontsov, E.V., Tokmashev, R.D., Guzina, B.B.: A physical perspective of the length scales in gradient elasticity through the prism of wave dispersion. Int. J. Solids Struct.50(22‐23), 3674-3684 (2013) |
| [57] | Kahrobaiyan, M., Asghari, M., Ahmadian, M.: Strain gradient beam element. Finite Elem. Anal. Des.68, 63-75 (2013) · Zbl 1302.74089 |
| [58] | Zhang, B., He, Y., Liu, D., Gan, Z., Shen, L.: Non‐classical Timoshenko beam element based on the strain gradient elasticity theory. Finite Elem. Anal. Des.79, 22-39 (2014) |
| [59] | Zhang, B., He, Y., Liu, D., Gan, Z., Shen, L.: A non‐classical Mindlin plate finite element based on a modified couple stress theory. Eur. J. Mech. A. Solids42, 63-80 (2013) · Zbl 1406.74458 |
| [60] | Kim, J., Reddy, J.N.: A general third‐order theory of functionally graded plates with modified couple stress effect and the von Kármán nonlinearity: theory and finite element analysis. Acta Mech.226(9), 2973-2998 (2015) · Zbl 1329.74171 |
| [61] | Kwon, Y.‐R., Lee, B.‐C.: Three dimensional elements with Lagrange multipliers for the modified couple stress theory. Comput. Mech.62(1), 97-110 (2018) · Zbl 1462.74157 |
| [62] | Kwon, Y.‐R., Lee, B.‐C.: A mixed element based on Lagrange multiplier method for modified couple stress theory. Comput. Mech.59(1), 117-128 (2017) · Zbl 1398.74352 |
| [63] | Torabi, J., Ansari, R., Darvizeh, M.: Application of a non‐conforming tetrahedral element in the context of the three‐dimensional strain gradient elasticity. Comput. Methods Appl. Mech. Eng.344, 1124-1143 (2019) · Zbl 1440.74445 |
| [64] | Babu, B., Patel, B.: A new computationally efficient finite element formulation for nanoplates using second‐order strain gradient Kirchhoff’s plate theory. Composites, Part B168, 302-311 (2019) |
| [65] | Sze, K., Wu, Z.: Twenty‐four-DOF four‐node quadrilateral elements for gradient elasticity. Int. J. Numer. Methods Biomed. Eng.119(2), 128-149 (2019) · Zbl 07845167 |
| [66] | Sze, K.Y., Yuan, W.C., Zhou, Y.X.: Four‐node tetrahedral elements for gradient‐elasticity analysis. Int. J. Numer. Methods Biomed. Eng. n/a (n/a) · Zbl 07863297 |
| [67] | Nguyen, H.X., Nguyen, T.N., Abdel‐Wahab, M., Bordas, S.P.A., Nguyen‐Xuan, H., Vo, T.P.: A refined quasi‐3D isogeometric analysis for functionally graded microplates based on the modified couple stress theory. Comput. Methods Appl. Mech. Eng.313, 904-940 (2017) · Zbl 1439.74453 |
| [68] | Thai, S., Thai, H.T., Vo, T.P., Patel, V.I.: Size‐dependant behaviour of functionally graded microplates based on the modified strain gradient elasticity theory and isogeometric analysis. Comput. Struct.190, 219-241 (2017) |
| [69] | Niiranen, J., Khakalo, S., Balobanov, V., Niemi, A.H.: Variational formulation and isogeometric analysis for fourth‐order boundary value problems of gradient‐elastic bar and plane strain/stress problems. Comput. Methods Appl. Mech. Eng.308, 182-211 (2016) · Zbl 1439.74036 |
| [70] | Niiranen, J., Niemi, A.H.: Variational formulations and general boundary conditions for sixth‐order boundary value problems of gradient‐elastic Kirchhoff plates. Eur. J. Mech. A. Solids61, 164-179 (2017) · Zbl 1406.74446 |
| [71] | Niiranen, J., Kiendl, J., Niemi, A.H., Reali, A.: Isogeometric analysis for sixth‐order boundary value problems of gradient‐elastic Kirchhoff plates. Comput. Methods Appl. Mech. Eng.316, 328-348 (2017) · Zbl 1439.74037 |
| [72] | Makvandi, R., Reiher, J.C., Bertram, A., Juhre, D.: Isogeometric analysis of first and second strain gradient elasticity. Comput. Mech.61(3), 351-363 (2018) · Zbl 1451.74218 |
| [73] | Balobanov, V., Niiranen, J.: Locking‐free variational formulations and isogeometric analysis for the Timoshenko beam models of strain gradient and classical elasticity, Comput. Methods Appl. Mech. Eng.339, 137-159 (2018) · Zbl 1440.74179 |
| [74] | Thai, C.H., Ferreira, A., Rabczuk, T., Nguyen‐Xuan, H.: Size‐dependent analysis of FG‐CNTRC microplates based on modified strain gradient elasticity theory. Eur. J. Mech. A. Solids72, 521-538 (2018) · Zbl 1406.74454 |
| [75] | Balobanov, V., Kiendl, J., Khakalo, S., Niiranen, J.: Kirchhoff-Love shells within strain gradient elasticity: Weak and strong formulations and an H 3 ‐conforming isogeometric implementation. Comput. Methods Appl. Mech. Eng.344(FEB.1), 837-857 (2019) · Zbl 1440.74057 |
| [76] | Bellman, R., Casti, J.: Differential quadrature and long‐term integration. J. Math. Anal. Appl.34(2), 235-238 (1971) · Zbl 0236.65020 |
| [77] | Bellman, R., Kashef, B., Casti, J.: Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J. Comput. Phys.10(1), 40-52 (1972) · Zbl 0247.65061 |
| [78] | Shu, C.: Differential Quadrature and its Application in Engineering, Springer Science & Business Media (2012) |
| [79] | Wang, X.: Differential Quadrature and Differential Quadrature Based Element Methods: Theory and Applications, Butterworth‐Heinemann (2015) · Zbl 1360.74003 |
| [80] | Wang, X.: Novel differential quadrature element method for vibration analysis of hybrid nonlocal Euler-Bernoulli beams. Appl. Math. Lett.77, 94-100 (2018) · Zbl 1469.74132 |
| [81] | Ishaquddin, M., Gopalakrishnan, S.: A novel weak form quadrature element for gradient elastic beam theories. Appl. Math. Modell.77, 1-16 (2020) · Zbl 1448.74106 |
| [82] | Ishaquddin, M., Gopalakrishnan, S.: Novel differential quadrature element method for higher order strain gradient elasticity theory, arXiv preprint arXiv:1802.08115 (2018) |
| [83] | Zhang, B., Li, H., Kong, L., Shen, H., Zhang, X.: Coupling effects of surface energy, strain gradient, and inertia gradient on the vibration behavior of small‐scale beams. Int. J. Mech. Sci.105834 (2020) |
| [84] | Zhang, B., Li, H., Kong, L., Wang, J., Shen, H.: Strain gradient differential quadrature beam finite elements. Comput. Struct.218, 170-189 (2019) |
| [85] | Zhang, B., Li, H., Kong, L., Shen, H., Zhang, X.: Size‐dependent vibration and stability of moderately thick functionally graded micro‐plates using a differential quadrature‐based geometric mapping scheme. Eng. Anal. Boundary Elem.108, 339-365 (2019) · Zbl 1464.74422 |
| [86] | Zhang, B., Li, H., Kong, L., Zhang, X., Shen, H.: Strain gradient differential quadrature Kirchhoff plate finite element with the C2 partial compatibility. Eur. J. Mech. A. Solids80, 103879 (2020) · Zbl 1472.74204 |
| [87] | Zhang, B., Li, H., Kong, L., Shen, H., Zhang, X.: Size‐dependent static and dynamic analysis of Reddy‐type micro‐beams by strain gradient differential quadrature finite element method. Thin Walled Struct.106496 (2019) |
| [88] | Zhang, B., Li, H., Kong, L., Zhang, X., Feng, Z.: Strain gradient differential quadrature finite element for moderately thick micro‐plates. Int. J. Numer. Methods Eng.n/a(n/a) (2020) |
| [89] | Zhang, B., Li, H., Liu, J., Shen, H., Zhang, X.: Surface energy‐enriched gradient elastic Kirchhoff plate model and a novel weak‐form solution scheme. Eur. J. Mech. A. Solids85, 104118 (2021) · Zbl 1473.74089 |
| [90] | Movassagh, A.A., Mahmoodi, M.J.: A micro‐scale modeling of Kirchhoff plate based on modified strain‐gradient elasticity theory. Eur. J. Mech. A. Solids40(40), 50-59 (2013) · Zbl 1406.74067 |
| [91] | Wang, B., Huang, S., Zhao, J., Zhou, S.: Reconsiderations on boundary conditions of Kirchhoff micro‐plate model based on a strain gradient elasticity theory. Appl. Math. Modell.40(15-16) (2016) 7303-7317 · Zbl 1471.74049 |
| [92] | Chakraverty, S.: Vibration of Plates, CRC press (2008) |
| [93] | Wang, X., Wang, Y.: Free vibration analysis of multiple‐stepped beams by the differential quadrature element method. Appl. Math. Comput.219(11), 5802-5810 (2013) · Zbl 1273.74135 |
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