Abstract
Previous studies have shown that Eringen’s differential nonlocal model would lead to the ill-posed mathematical formulation for axisymmetric bending of circular microplates. Based on the nonlocal integral models along the radial and circumferential directions, we propose nonlocal integral polar models in this work. The proposed strain-and stress-driven two-phase nonlocal integral polar models are applied to model the axisymmetric bending of circular microplates. The governing differential equations and boundary conditions (BCs) as well as constitutive constraints are deduced. It is found that the purely strain-driven nonlocal integral polar model turns to a traditional nonlocal differential polar model if the constitutive constraints are neglected. Meanwhile, the purely strain- and stress-driven nonlocal integral polar models are ill-posed, because the total number of the differential orders of the governing equations is less than that of the BCs plus constitutive constraints. Several nominal variables are introduced to simplify the mathematical expression, and the general differential quadrature method (GDQM) is applied to obtain the numerical solutions. The results from the current models (CMs) are compared with the data in the literature. It is clearly established that the consistent softening and toughening effects can be obtained for the strain- and stress-driven local/nonlocal integral polar models, respectively. The proposed two-phase local/nonlocal integral polar models (TPNIPMs) may provide an efficient method to design and optimize the plate-like structures for microelectro-mechanical systems.
Similar content being viewed by others
References
WANG, B., ZHOU, S., ZHAO, J., and CHEN, X. Pull-in instability of circular plate MEMS: a new model based on strain gradient elasticity theory. International Journal of Applied Mechanics, 4(1), 1250003 (2012)
CARUNTU, D. I. and OYERVIDES, R. Voltage response of primary resonance of electrostatically actuated MEMS clamped circular plate resonators. Journal of Computational Nonlinear Dynamics, 11(4), 041021 (2016)
YANG, W. D., KANG, W. B., and WANG, X. Thermal and surface effects on the pull-in characteristics of circular nanoplate NEMS actuator based on nonlocal elasticity theory. Applied Mathematical Modelling, 43, 321–336 (2017)
SAADATMAND, M. and KOOK, J. Multi-objective optimization of a circular dual back-plate MEMS microphone: tradeoff between pull-in voltage, sensitivity and resonance frequency. Microsystem Technologies, 25(8), 2937–2947 (2019)
KOITER, W. T. Couple stresses in the theory of elasticity, i and ii. Koninklijke Nederlandse Akademie van Wetenschappen Proceedings Series B, 67, 17–44 (1964)
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)
MINDLIN, R. D. and ESHEL, N. N. On first strain-gradient theories in linear elasticity. International Journal of Solids and Structures, 4(1), 109–124 (1968)
MINDLIN, R. D. Second gradient of strain and surface-tension in linear elasticity. International Journal of Solids and Structures, 1(4), 417–438 (1965)
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)
ERINGEN, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54(9), 4703 (1983)
ERINGEN, A. C. Theory of nonlocal elasticity and some applications. Res Mechanica, 21(4), 313–342 (1987)
ZHANG, P. and QING, H. Two-phase nonlocal integral models with a bi-Helmholtz averaging kernel for nanorods. Applied Mathematics and Mechanics (English Edition), 42(10), 1379–1396 (2021) https://doi.org/10.1007/s10483-021-2774-9
ZHANG, P. and QING, H. A bi-Helmholtz type of two-phase nonlocal integral model for buckling of Bernoulli-Euler beams under non-uniform temperature. Journal of Thermal Stresses, 44(9), 1053–1067 (2021)
ZHANG, P. and QING, H. Closed-form solution in bi-Helmholtz kernel based two-phase nonlocal integral models for functionally graded Timoshenko beams. Composite Structures, 265, 113770 (2021)
KRÖNER, E. Elasticity theory of materials with long range cohesive forces. International Journal of Solids and Structures, 3, 731–742 (1967)
ERINGEN, A. C. Linear theory of nonlocal elasticity and dispersion of plane waves. International Journal of Engineering Science, 10(5), 425–435 (1972)
ERINGEN, A. C. and EDELEN, D. G. B. On nonlocal elasticity. International Journal of Engineering Science, 10(3), 233–248 (1972)
LU, L., ZHU, L., GUO, X., ZHAO, J., and LIU, G. A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells. Applied Mathematics and Mechanics (English Edition), 40(12), 1695–1722 (2019) https://doi.org/10.1007/s10483-019-2549-7
PENG, W., CHEN, L., and HE, T. Nonlocal thermoelastic analysis of a functionally graded material microbeam. Applied Mathematics and Mechanics (English Edition), 42(6), 855–870 (2021) https://doi.org/10.1007/s10483-021-2742-9
LI, X., LI, L., HU, Y., DING, Z., and DENG, W. Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory. Composite Structures, 165, 250–265 (2017)
DUAN, W. H. and WANG, C. M. Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory. Nanotechnology, 18(38), 385704 (2007)
ARTAN, R. and LEHMANN, L. Initial values method for symmetric bending of micro/nano annular circular plates based on nonlocal plate theory. Journal of Computational Theoretical Nanoscience, 6(5), 1125–1130 (2009)
YU, Y. M. and LIM, C. W. Nonlinear constitutive model for axisymmetric bending of annular graphene-like nanoplate with gradient elasticity enhancement effects. Journal of Engineering Mechanics, 139(8), 1025–1035 (2013)
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(1), 61 (2019)
PEDDIESON, J., BUCHANAN, G., and MCNITT, R. Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science, 41(3–5), 305–312 (2003)
BENVENUTI, E. and SIMONE, A. One-dimensional nonlocal and gradient elasticity: closed-form solution and size effect. Mechanics Research Communications, 48, 46–51 (2013)
FERNANDEZ-SAEZ, J., ZAERA, R., LOYA, J. A., and REDDY, J. N. Bending of Euler-Bernoulli beams using Eringen’s integral formulation: a paradox resolved. International Journal of Engineering Science, 99, 107–116 (2016)
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)
ZHANG, J. Q., QING, H., and GAO, C. F. Exact and asymptotic bending analysis ofmicrobeams under different boundary conditions using stress-derived nonlocal integral model. Zeitschrift für Angewandte Mathematik und Mechanik, 100(1), 201900148 (2020)
ZHANG, P., QING, H., and GAO, C. Theoretical analysis for static bending of circular Euler-Bernoulli beam using local and Eringen’s nonlocal integral mixed model. Zeitschrift für Angewandte Mathematik und Mechanik, 99(8), 201800329 (2019)
ZHANG, P., QING, H., and GAO, C. F. Analytical solutions of static bending of curved Timoshenko microbeams using Eringen’s two-phase local/non-local integral model. Zeitschrift für Angewandte Mathematik und Mechanik, 100(7), 201900207 (2020)
WANG, Y. B., ZHU, X. W., and DAI, H. H. Exact solutions for the static bending of Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model. AIP Advances, 6(8), 085114 (2016)
ROMANO, G., BARRETTA, R., DIACO, M., and DE SCIARRA, F. M. Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. International Journal of Mechanical Sciences, 121, 151–156 (2017)
ROMANO, G., LUCIANO, R., BARRETTA, R., and DIACO, M. Nonlocal integral elasticity in nanostructures, mixtures, boundary effects and limit behaviours. Continuum Mechanics and Thermodynamics, 30(3), 641–655 (2018)
ROMANO, G. and BARRETTA, R. Nonlocal elasticity in nanobeams: the stress-driven integral model. International Journal of Engineering Science, 115, 14–27 (2017)
BARRETTA, R., CAPORALE, A., FAGHIDIAN, S. A., LUCIANO, R., DE SCIARRA, F. M., and MEDAGLIA, C. M. A stress-driven local-nonlocal mixture model for Timoshenko nano-beams. Composites Part B-Engineering, 164, 590–598 (2019)
BARRETTA, R., CANADIJA, M., LUCIANO, R., and DE SCIARRA, F. M. Stress-driven modeling of nonlocal thermoelastic behavior of nanobeams. International Journal of Engineering Science, 126, 53–67 (2018)
ZHANG, P., QING, H., and GAO, C. F. Exact solutions for bending of Timoshenko curved nanobeams made of functionally graded materials based on stress-driven nonlocal integral model. Composite Structures, 245, 112362 (2020)
REN, Y. M. and QING, H. Bending and buckling analysis of functionally graded Euler-Bernoulli beam using stress-driven nonlocal integral model with bi-Helmholtz kernel. International Journal of Applied Mechanics, 13(4), 2150041 (2021)
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)
SHARIATI, M., SHISHESAZ, M., MOSALMANI, R. S., and ROKNIZADEH, S. A. Size effect on the axisymmetric vibrational response of functionally graded circular nano-plate based on the nonlocal stress-driven method. Journal of Applied and Computational Mechanics, 8(3), 962–980 (2022)
ZHANG, P. and QING, H. Well-posed two-phase nonlocal integral models for free vibration of nanobeams in context with higher-order refined shear deformation theory. Journal of Vibration and Control (2021) https://doi.org/10.1177/10775463211039902
ZHANG, P. and QING, H. On well-posedness of two-phase nonlocal integral models for higher-order refined shear deformation beams. Applied Mathematics and Mechanics (English Edition), 42(7), 931–950 (2021) https://doi.org/10.1007/s10483-021-2750-8
MCDOWELL, M. T., LEACH, A. M., and GAILL, K. On the elastic modulus of metallic nanowires. Nano Letter, 8(11), 3613–3618 (2008)
FEDORCHENKO, A. I., WANG, A. B., and CHENG, H. H. Thickness dependence of nanofilm elastic modulus. Applied Physics Letters, 94(15), 152111 (2009)
WU, T. and LIU, G. The generalized differential quadrature rule for fourth-order differential equations. International Journal for Numerical Methods in Engineering, 50(8), 1907–1929 (2001)
CHEN, C. The Timoshenko beam model of the differential quadrature element method. Computational Mechanics, 24(1), 65–69 (1999)
REDDY, J. N. Theory and Analysis of Elastic Plates and Shells, CRC Press, New York (2016)
Funding
Project supported by the National Natural Science Foundation of China (No. 12172169) and the Research Fund of State Key Laboratory of Mechanics and the Priority Academic Program Development of Jiangsu Higher Education Institutions of China
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Qing, H. Well-posedness of two-phase local/nonlocal integral polar models for consistent axisymmetric bending of circular microplates. Appl. Math. Mech.-Engl. Ed. 43, 637–652 (2022). https://doi.org/10.1007/s10483-022-2843-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-022-2843-9
Key words
- softening effect
- toughening effect
- circular microplate
- nonlocal integral model
- general differential quadrature method (GDQM)