Size-dependent axisymmetric bending and buckling analysis of functionally graded sandwich Kirchhoff nanoplates using nonlocal strain gradient integral model

doi:10.1007/s10483-025-3222-9

Abstract

Abstract:

This paper extends the one-dimensional (1D) nonlocal strain gradient integral model (NStraGIM) to the two-dimensional (2D) Kirchhoff axisymmetric nanoplates, based on nonlocal strain gradient integral relations formulated along both the radial and circumferential directions. By transforming the proposed integral constitutive equations into the equivalent differential forms, complemented by the corresponding constitutive boundary conditions (CBCs), a well-posed mathematical formulation is established for analyzing the axisymmetric bending and buckling of annular/circular functionally graded (FG) sandwich nanoplates. The boundary conditions at the inner edge of a solid nanoplate are derived by L'Hôspital's rule. The numerical solution is obtained by the generalized differential quadrature method (GDQM). The accuracy of the proposed model is validated through comparison with the data from the existing literature. A parameter study is conducted to demonstrate the effects of FG sandwich parameters, size parameters, and nonlocal gradient parameters.

Key words: size effect, nonlocal strain gradient integral model (NStraGIM), bending, buckling, Kirchhoff annular/circular nanoplate, functionally graded (FG) sandwich material

2010 MSC Number:

O343.1

Chang LI, Hai QING. Size-dependent axisymmetric bending and buckling analysis of functionally graded sandwich Kirchhoff nanoplates using nonlocal strain gradient integral model. Applied Mathematics and Mechanics (English Edition), 2025, 46(3): 467-484.

TrendMD

Figures/Tables 15

Fig. 1

Table 1

Different types of volume compositions"

Integral array	1-8-1	1-2-1	1-1-1	2-1-2
Volume fraction of the core	0.8	0.5	1/3	0.2
h₂	$− 2 h / 5$	$− h / 4$	$− h / 6$	$− h / 10$
h₃	$2 h / 5$	$h / 4$	$h / 6$	$h / 10$

Table 1

Table 2

Table 3

The NBL of an annular nanoplate (κ, c→0+)"

Condition	Method	CC	SS	CS	SC
$a / (b − a) = 0.1$	NStraGIM	41.121	14.632	18.363	33.024
$a / (b − a) = 0.1$	Reddy^[58]	41.192	14.636	18.344	33.046
$a / (b − a) = 0.5$	NStraGIM	39.714	10.841	18.619	23.878
$a / (b − a) = 0.5$	Reddy^[58]	39.868	10.795	18.609	23.841

Table 3

Table 4

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Table 5

Fig. 8

Fig. 9

Table 6

NBLs of local FG sandwich annular and circular nanoplates under different BCs with the power-law index α=0.5 and the volume composition set as 1-2-1"

Aspect ratio	Annular nanoplate				Circular nanoplate
Aspect ratio	CC	CS	SC	SS	Clamped	Simply-supported
$a / (b − a) = 0.1$	28.327 0	12.734 5	22.942 4	10.240 7	10.531 8	3.014 6
$a / (b − a) = 0.5$	27.374 2	12.897 8	16.600 5	7.587 8	10.531 8	3.014 6

Table 6

References 43

[44]	ZHAO, L., WEI, P., and LI, Y. Free vibration of thermo-elastic microplate based on spatiotemporal fractional-order derivatives with nonlocal characteristic length and time. Applied Mathematics and Mechanics (English Edition), 44(1), 109–124 (2023) https://doi.org/10.1007/s10483-023-2933-8
[45]	ÖZMEN, R. Thermomechanical vibration and buckling response of magneto-electro-elastic higher order laminated nanoplates. Applied Mathematical Modelling, 122, 373–400 (2023)
[46]	ZAERA, R., SERRANO, O., and FERNANDEZ-SAEZ, J. On the consistency of the nonlocal strain gradient elasticity. International Journal of Engineering Science, 138, 65–81 (2019)
[47]	LI, C., QING, H., and GAO, C. Theoretical analysis for static bending of Euler-Bernoulli beam using different nonlocal gradient models. Mechanics of Advanced Materials and Structures, 28(19), 1965–1977 (2021)
[48]	BIAN, P. L. and QING, H. On bending consistency of Timoshenko beam using differential and integral nonlocal strain gradient models. Zeitschrift für Angewandte Mathematik und Mechanik, 101(8), e202000132 (2021)
[49]	BARRETTA, R. and DE SCIARRA, F. M. Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams. International Journal of Engineering Science, 130, 187–198 (2018)
[50]	NGUYEN, T. K., VO, T. P., NGUYEN, B. D., and LEE, J. An analytical solution for buckling and vibration analysis of functionally graded sandwich beams using a quasi-3D shear deformation theory. Composite Structures, 156, 238–252 (2016)
[51]	LIU, H., LV, Z., and WU, H. Nonlinear free vibration of geometrically imperfect functionally graded sandwich nanobeams based on nonlocal strain gradient theory. Composite Structures, 214, 47–61 (2019)
[52]	BARRETTA, R. and MAROTTI DE SCIARRA, F. Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams. International Journal of Engineering Science, 130, 187–198 (2018)
[53]	WANG, X. Novel differential quadrature element method for vibration analysis of hybrid nonlocal Euler-Bernoulli beams. Applied Mathematics Letters, 77, 94–100 (2018)
[1]	FOX, C. H. J., CHEN, X., and MCWILLIAM, S. Analysis of the deflection of a circular plate with an annular piezoelectric actuator. Sensors and Actuators A: Physical, 133(1), 180–194 (2007)
[2]	TYLIKOWSKI, A. Influence of bonding layer on piezoelectric actuators of an axisymmetrical annular plate. Journal of Theoretical and Applied Mechanics, 38(3), 607–621 (2000)
[3]	DONOSO, A. and CARLOS BELLIDO, J. Distributed piezoelectric modal sensors for circular plates. Journal of Sound and Vibration, 319(1-2), 50–57 (2009)
[4]	YANG, L., LI, P., FANG, Y., and GE, X. A generalized methodology for thermoelastic damping in axisymmetric vibration of circular plate resonators covered by multiple partial coatings. Thin-Walled Structures, 162, 107576 (2021)
[5]	MA, C., CHEN, S., and GUO, F. Thermoelastic damping in micromechanical circular plate resonators with radial pre-tension. Journal of Thermal Stresses, 43(2), 175–190 (2020)
[6]	LAM, D. C. C., YANG, F., CHONG, A. C. M., WANG, J., and TONG, P. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids, 51(8), 1477–1508 (2003)
[7]	CHEN, Y. X., DORGAN, B. L., MCILROY, D. N., and ASTON, D. E. On the importance of boundary conditions on nanomechanical bending behavior and elastic modulus determination of silver nanowires. Journal of Applied Physics, 100(10), 104301 (2006)
[8]	MCDOWELL, M. T., LEACH, A. M., and GAILL, K. On The elastic modulus of metallic nanowires. Nano Letters, 8(11), 3613–3618 (2008)
[9]	MOTZ, C., WEYGAND, D., SENGER, J., and GUMBSCH, P. Micro-bending tests: a comparison between three-dimensional discrete dislocation dynamics simulations and experiments. Acta Materialia, 56(9), 1942–1955 (2008)
[10]	SUN, C. T. and ZHANG, H. T. Size-dependent elastic moduli of platelike nanomaterials. Journal of Applied Physics, 93(2), 1212–1218 (2003)
[11]	FEDORCHENKO, A. I., WANG, A. B., and CHENG, H. H. Thickness dependence of nanofilm elastic modulus. Applied Physics Letters, 94(15), 152111 (2009)
[12]	CAI, J., LI, Y. L., MO, D., and WANG, Y. D. Softening effect on elastic moduli of Fe, Nb, Cu, and RuAl nanoparticles. Journal of Nanoscience and Nanotechnology, 19(12), 7899–7905 (2019)
[13]	YAN, J. W., TONG, L. H., LUO, R. J., and GAO, D. Thickness of monolayer h-BN nanosheet and edge effect on free vibration behaviors. International Journal of Mechanical Sciences, 164, 105163 (2019)
[14]	YAN, J. W. and ZHANG, W. An atomistic-continuum multiscale approach to determine the exact thickness and bending rigidity of monolayer graphene. Journal of Sound and Vibration, 514, 116464 (2021)
[15]	YAN, J. W., ZHU, J. H., LI, C., ZHAO, X. S., and LIM, C. W. Decoupling the effects of material thickness and size scale on the transverse free vibration of BNNTs based on beam models. Mechanical Systems and Signal Processing, 166, 108440 (2022)
[16]	YAN, J., YI, S., and YUAN, X. Graphene and its composites: a review of recent advances and applications in logistics transportation. Packaging Technology and Science, 37(4), 335–361 (2024)
[17]	MINDLIN, R. D. Second gradient of strain and surface-tension in linear elasticity. International Journal of Solids and Structures, 1(4), 417–438 (1965)
[18]	MINDLIN, R. D. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 16, 51–78 (1964)
[19]	SAHMANI, S. and SAFAEI, B. Nonlinear free vibrations of bi-directional functionally graded micro/nano-beams including nonlocal stress and microstructural strain gradient size effects. Thin-Walled Structures, 140, 342–356 (2019)
[20]	JIN, J., HU, N., and HU, H. Size effects on the mixed modes and defect modes for a nano-scale phononic crystal slab. Applied Mathematics and Mechanics (English Edition), 44(1), 21–34 (2023) https://doi.org/10.1007/s10483-023-2945-6
[21]	YANG, F., CHONG, A. C. M., LAM, D. C. C., and TONG, P. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39(10), 2731–2743 (2002)
[22]	KARAMI, B., GHAYESH, M. H., and FANTUZZI, N. Quasi-3D free and forced vibrations of poroelastic microplates in the framework of modified couple stress theory. Composite Structures, 330, 117840 (2024)
[23]	ERINGEN, A. C. Theory of nonlocal elasticity and some applications. Res Mechanica, 21(4), 313–342 (1987)
[24]	ERINGEN, A. C. and EDELEN, D. G. B. On nonlocal elasticity. International Journal of Engineering Science, 10(3), 233–248 (1972)
[25]	ROMANO, G. and BARRETTA, R. Nonlocal elasticity in nanobeams: the stress-driven integral model. International Journal of Engineering Science, 115, 14–27 (2017)
[26]	TORKAMAN-ASADI, M. A., RAHMANIAN, M., and FIROUZ-ABADI, R. D. Free vibrations and stability of high-speed rotating carbon nanotubes partially resting on Winkler foundations. Composite Structures, 126, 52–61 (2015)
[27]	BARRETTA, R., FAZELZADEH, S. A., FEO, L., GHAVANLOO, E., and LUCIANO, R. Nonlocal inflected nano-beams: a stress-driven approach of bi-Helmholtz type. Composite Structures, 200, 239–245 (2018)
[28]	TANG, Y., BIAN, P. L., and QING, H. Buckling and vibration analysis of axially functionally graded nanobeam based on local stress- and strain-driven two-phase local/nonlocal integral models. Thin-Walled Structures, 202, 112162 (2024)
[29]	LIM, C. W. On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection. Applied Mathematics and Mechanics (English Edition), 31(1), 37–54 (2010) https://doi.org/10.1007/s10483-010-0105-7
[30]	YAN, J. W., TONG, L. H., LI, C., ZHU, Y., and WANG, Z. W. Exact solutions of bending deflections for nano-beams and nano-plates based on nonlocal elasticity theory. Composite Structures, 125, 304–313 (2015)
[31]	LI, C., YAO, L., CHEN, W., and LI, S. Comments on nonlocal effects in nano-cantilever beams. International Journal of Engineering Science, 87, 47–57 (2015)
[32]	SUN, J., WANG, Z., ZHOU, Z., XU, X., and LIM, C. W. Surface effects on the buckling behaviors of piezoelectric cylindrical nanoshells using nonlocal continuum model. Applied Mathematical Modelling, 59, 341–356 (2018)
[33]	SAHMANI, S. and ANSARI, R. On the free vibration response of functionally graded higher-order shear deformable microplates based on the strain gradient elasticity theory. Composite Structures, 95, 430–442 (2013)
[34]	ANSARI, R., GHOLAMI, R., FAGHIH SHOJAEI, M., MOHAMMADI, V., and SAHMANI, S. Bending, buckling and free vibration analysis of size-dependent functionally graded circular/annular microplates based on the modified strain gradient elasticity theory. European Journal of Mechanics-A/Solids, 49, 251–267 (2015)
[35]	REDDY, J. N., ROMANOFF, J., and LOYA, J. A. Nonlinear finite element analysis of functionally graded circular plates with modified couple stress theory. European Journal of Mechanics-A/Solids, 56, 92–104 (2016)
[36]	REDDY, J. N. and BERRY, J. Nonlinear theories of axisymmetric bending of functionally graded circular plates with modified couple stress. Composite Structures, 94(12), 3664–3668 (2012)
[37]	WEI, L. and QING, H. Bending, buckling and vibration analysis of bi-directional functionally graded circular/annular microplate based on MCST. Composite Structures, 292, 115633 (2022)
[54]	JIN, C. and WANG, X. Quadrature element method for vibration analysis of functionally graded beams. Engineering Computations, 34(4), 1293–1313 (2017)
[55]	AL-SHUJAIRI, M. and MOLLAMAHMUTOĞLU, Ç. Buckling and free vibration analysis of functionally graded sandwich micro-beams resting on elastic foundation by using nonlocal strain gradient theory in conjunction with higher order shear theories under thermal effect. Composites Part B: Engineering, 154, 292–312 (2018)
[56]	JIN, C. and WANG, X. Accurate free vibration analysis of Euler functionally graded beams by the weak form quadrature element method. Composite Structures, 125, 41–50 (2015)
[57]	TORNABENE, F., FANTUZZI, N., UBERTINI, F., and VIOLA, E. Strong formulation finite element method based on differential quadrature: a survey. Applied Mechanics Reviews, 67(2), 020801 (2015)
[58]	REDDY, J. N. Theory and Analysis of Elastic Plates and Shells, 2nd ed., CRC Press, New York (2006)
[38]	YUKSELER, R. F. Exact nonlocal solutions of circular nanoplates subjected to uniformly distributed loads and nonlocal concentrated forces. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 42, 61 (2020)
[39]	BARRETTA, R., FAGHIDIAN, S. A., and DE SCIARRA, F. M. Stress-driven nonlocal integral elasticity for axisymmetric nano-plates. International Journal of Engineering Science, 136, 38–52 (2019)
[40]	LI, S., ZHENG, W., and LI, L. Spatiotemporally nonlocal homogenization method for viscoelastic porous metamaterial structures. International Journal of Mechanical Sciences, 282, 109572 (2024)
[41]	LI, S., XU, E., ZHAN, X., ZHENG, W., and LI, L. Stress-driven nonlocal homogenization method for cellular structures. Aerospace Science and Technology, 155, 109632 (2024)
[42]	LI, R., LI, L., and JIANG, Y. A physics-based nonlocal theory for particle-reinforced polymer composites. International Journal of Mechanical Sciences, 285, 109800 (2025)
[43]	LIM, C. W., ZHANG, G., and REDDY, J. N. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids, 78, 298–313 (2015)

Method	Clamped end	Simply supported
NStraGIM	0.015 620	0.065 640
Reddy^[58]	0.015 625	0.065 625

Method	Clamped end	Simply supported
NStraGIM	14.570	4.196
Reddy^[58]	14.682	4.198

Type	BC	Volume composition	FG index α
Type	BC	Volume composition	0	0.5	1	2	3	5	10
Annular plate	CC	1-8-1	32.104 390	27.464 940	25.242 500	23.115 830	22.099 640	21.123 120	20.276 570
		1-1-1	32.104 390	20.519 630	15.614 120	11.540 060	9.891 029	8.546 865	7.616 580
		1-0-1	32.104 390	17.886 730	12.461 570	8.533 008	7.223 487	6.381 652	6.005 541
	CS	1-8-1	16.821 740	14.390 810	13.226 310	12.112 010	11.579 550	11.067 880	10.624 320
		1-1-1	16.821 740	10.751 670	8.181 333	6.046 644	5.182 604	4.478 302	3.990 861
		1-0-1	16.821 740	9.372 112	6.529 491	4.471 041	3.784 891	3.343 795	3.146 724
	SC	1-8-1	20.444 190	17.489 770	16.074 510	14.720 250	14.073 130	13.451 280	12.912 190
		1-1-1	20.444 190	13.066 970	9.943 129	7.348 750	6.298 644	5.442 675	4.850 266
		1-0-1	20.444 190	11.390 330	7.935 574	5.433 850	4.599 943	4.063 859	3.824 350
	SS	1-8-1	10.532 120	9.295 863	8.710 415	8.156 260	7.894 183	7.644 385	7.429 746
		1-1-1	10.532 120	7.491 174	6.296 477	5.451 963	5.205 139	5.089 304	5.082 384
		1-0-1	10.532 120	6.836 203	5.621 308	5.088 682	5.104 743	5.218 890	5.304 845
Circular plate	Clamped	1-8-1	11.920 310	10.197 690	9.372 499	8.582 873	8.205 564	7.842 980	7.528 660
		1-1-1	11.920 310	7.618 906	5.797 499	4.284 805	3.672 524	3.173 438	2.828 025
		1-0-1	11.920 310	7.163 657	5.229 878	3.705 083	3.129 395	2.695 962	2.432 498
	Simply-supported	1-8-1	4.007 257	3.428 163	3.150 759	2.885 309	2.758 469	2.636 579	2.530 914
		1-1-1	4.007 257	2.561 252	1.948 949	1.440 425	1.234 594	1.066 816	0.950 699
		1-0-1	4.007 257	2.232 615	1.555 448	1.065 087	0.901 633	0.796 555	0.749 609