Variational Formulations and Isogeometric Analysis of Timoshenko–Ehrenfest Microbeam Using a Reformulated Strain Gradient Elasticity Theory
Abstract
:1. Introduction
2. Reformulated Strain Gradient Elasticity Theory
3. Non-Classical Timoshenko–Ehrenfest Beam Model
4. Isogeometric Analysis Approach
4.1. NURBS Basis Functions
4.2. Discretized Equations
4.3. Static Bending
4.4. Free Vibration
5. Examples and Discussions
5.1. Static Bending Problem
5.2. Free Vibration Problem
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Lam, D.C.; Yang, F.; Chong, A.C.M.; Wang, J.; Tong, P. Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 2003, 51, 1477–1508. [Google Scholar] [CrossRef]
- McFarland, A.W.; Colton, J.S. Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J. Micromech. Microeng. 2005, 15, 1060. [Google Scholar] [CrossRef]
- Maranganti, R.; Sharma, P. Length scales at which classical elasticity breaks down for various materials. Phys. Rev. Lett. 2007, 98, 195504. [Google Scholar] [CrossRef] [Green Version]
- Gao, X.-L. An expanding cavity model incorporating strain-hardening and indentation size effects. Int. J. Solids Struct. 2006, 43, 6615–6629. [Google Scholar] [CrossRef] [Green Version]
- Mindlin, R.D. Microstructure in Linear Elasticity; Columbia University New York Departments of Civil Engineering and Engineering Mechanics: New York, NY, USA, 1963. [Google Scholar]
- Eringen, A.C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 1983, 54, 4703–4710. [Google Scholar] [CrossRef]
- Eringen, A.C.; Wegner, J. Nonlocal continuum field theories. Appl. Mech. Rev. 2003, 56, B20–B22. [Google Scholar] [CrossRef]
- Toupin, R. Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 1962, 11, 385–414. [Google Scholar] [CrossRef] [Green Version]
- Mindlin, R. Influence of Couple-Stresses on Stress Concentrations; Columbia University New York: New York, NY, USA, 1962. [Google Scholar]
- Koiter, W. Couple stresses in the theory of elasticity, I & II. Proc. K. Ned. Akad. Wet. 1964, B, 17–44. [Google Scholar]
- Neff, P.; Forest, S. A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. J. Elast. 2007, 87, 239–276. [Google Scholar] [CrossRef]
- Eringen, A.C.; Suhubi, E.S. Nonlinear theory of simple microelastic solids. Int. J. Eng. Sci. 1964, 2, 389–404. [Google Scholar] [CrossRef]
- Eringen, A.C. Microcontinuum Field Theories: I. Foundations and Solids; Springer Science & Business Media: Heidelberg, Germany, 2012. [Google Scholar]
- Zhang, G.Y.; Gao, X.-L.; Zheng, C.Y.; Mi, C.W. A non-classical Bernoulli-Euler beam model based on a simplified micromorphic elasticity theory. Mech. Mater. 2021, 161, 103967. [Google Scholar] [CrossRef]
- Mindlin, R.D.; Eshel, N. On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 1968, 4, 109–124. [Google Scholar] [CrossRef]
- Polizzotto, C. A hierarchy of simplified constitutive models within isotropic strain gradient elasticity. Eur. J. Mech.-A Solids 2017, 61, 92–109. [Google Scholar] [CrossRef]
- Hutchinson, J.; Fleck, N. Strain gradient plasticity. Adv. Appl. Mech. 1997, 33, 295–361. [Google Scholar]
- Sedighi, H.M.; Koochi, A.; Abadyan, M. Modeling the size dependent static and dynamic pull-in instability of cantilever nanoactuator based on strain gradient theory. Int. J. Appl. Mech. 2014, 6, 1450055. [Google Scholar] [CrossRef]
- 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. 2009, 47, 487–498. [Google Scholar] [CrossRef]
- Abbasi, M. Size dependent vibration behavior of an AFM with sidewall and top-surface probes based on the strain gradient elasticity theory. Int. J. Appl. Mech. 2015, 7, 1550046. [Google Scholar] [CrossRef]
- Wang, Y.Q.; Zhao, H.L.; Ye, C.; Zu, J.W. A porous microbeam model for bending and vibration analysis based on the sinusoidal beam theory and modified strain gradient theory. Int. J. Appl. Mech. 2018, 10, 1850059. [Google Scholar] [CrossRef]
- Khakalo, S.; Niiranen, J. Form II of Mindlin’s second strain gradient theory of elasticity with a simplification: For materials and structures from nano- to macro-scales. Eur. J. Mech.-A Solids 2018, 71, 292–319. [Google Scholar] [CrossRef]
- Altan, B.; Aifantis, E. On some aspects in the special theory of gradient elasticity. J. Mech. Behav. Mater. 1997, 8, 231–282. [Google Scholar] [CrossRef]
- Gao, X.-L.; Park, S. Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem. Int. J. Solids Struct. 2007, 44, 7486–7499. [Google Scholar] [CrossRef]
- Papargyri-Beskou, S.; Polyzos, D.; Beskos, D. Wave dispersion in gradient elastic solids and structures: A unified treatment. Int. J. Solids Struct. 2009, 46, 3751–3759. [Google Scholar] [CrossRef] [Green Version]
- Gourgiotis, P.; Georgiadis, H. Plane-strain crack problems in microstructured solids governed by dipolar gradient elasticity. J. Mech. Phys. Solids 2009, 57, 1898–1920. [Google Scholar] [CrossRef] [Green Version]
- Liang, X.; Hu, S.; Shen, S. A new Bernoulli–Euler beam model based on a simplified strain gradient elasticity theory and its applications. Compos. Struct. 2014, 111, 317–323. [Google Scholar] [CrossRef]
- Lazopoulos, K. Post-buckling problems for long elastic beams. Acta Mech. 2003, 164, 189–198. [Google Scholar] [CrossRef]
- Khakalo, S.; Balobanov, V.; Niiranen, J. Modelling size-dependent bending, buckling and vibrations of 2D triangular lattices by strain gradient elasticity models: Applications to sandwich beams and auxetics. Int. J. Eng. Sci. 2018, 127, 33–52. [Google Scholar] [CrossRef] [Green Version]
- Ansari, R.; Torabi, J. Numerical study on the free vibration of carbon nanocones resting on elastic foundation using nonlocal shell model. Appl. Phys. A 2016, 122, 1073. [Google Scholar] [CrossRef]
- Ansari, R.; Torabi, J. Nonlocal vibration analysis of circular double-layered graphene sheets resting on an elastic foundation subjected to thermal loading. Acta Mech. Sin. 2016, 32, 841–853. [Google Scholar] [CrossRef]
- Torabi, J.; Ansari, R.; Zabihi, A.; Hosseini, K. Dynamic and pull-in instability analyses of functionally graded nanoplates via nonlocal strain gradient theory. Mech. Based Des. Struct. Mach. 2020, 50, 588–608. [Google Scholar]
- Zabihi, A.; Ansari, R.; Torabi, J.; Samadani, F.; Hosseini, K. An analytical treatment for pull-in instability of circular nanoplates based on the nonlocal strain gradient theory with clamped boundary condition. Mater. Res. Express 2019, 6, 0950b3. [Google Scholar] [CrossRef]
- Yang, F.A.C.M.; Chong, A.C.M.; Lam, D.C.C.; Tong, P. Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 2002, 39, 2731–2743. [Google Scholar] [CrossRef]
- Park, S.; Gao, X.-L. Variational formulation of a modified couple stress theory and its application to a simple shear problem. Z. Angew. Math. Phys. 2008, 59, 904–917. [Google Scholar] [CrossRef]
- Park, S.; Gao, X. Bernoulli–Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 2006, 16, 2355. [Google Scholar] [CrossRef]
- Gao, X.-L.; Mahmoud, F. A new Bernoulli–Euler beam model incorporating microstructure and surface energy effects. Z. Angew. Math. Phys. 2014, 65, 393–404. [Google Scholar] [CrossRef]
- Ma, H.; Gao, X.-L.; Reddy, J. A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 2008, 56, 3379–3391. [Google Scholar] [CrossRef]
- Gao, X.-L. A new Timoshenko beam model incorporating microstructure and surface energy effects. Acta Mech. 2015, 226, 457–474. [Google Scholar] [CrossRef]
- Ma, H.; Gao, X.-L.; Reddy, J. A nonclassical Reddy-Levinson beam model based on a modified couple stress theory. Int. J. Multiscale Comput. Eng. 2010, 8, 167–180. [Google Scholar] [CrossRef]
- Gao, X.-L.; Zhang, G. A microstructure-and surface energy-dependent third-order shear deformation beam model. Z. Angew. Math. Phys. 2015, 66, 1871–1894. [Google Scholar] [CrossRef]
- Hong, J.; Wang, S.; Zhang, G.; Mi, C. Bending, buckling and vibration analysis of complete microstructure-dependent functionally graded material microbeams. Int. J. Appl. Mech. 2021, 13, 2150057. [Google Scholar] [CrossRef]
- Hong, J.; Wang, S.; Zhang, G.; Mi, C. On the Bending and Vibration Analysis of Functionally Graded Magneto-Electro-Elastic Timoshenko Microbeams. Crystals 2021, 11, 1206. [Google Scholar] [CrossRef]
- Zhang, G.; Gao, X.-L. A new Bernoulli–Euler beam model based on a reformulated strain gradient elasticity theory. Math. Mech. Solids 2020, 25, 630–643. [Google Scholar] [CrossRef]
- Hughes, T.J.; Cottrell, J.A.; Bazilevs, Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 2005, 194, 4135–4195. [Google Scholar] [CrossRef] [Green Version]
- Borković, A.; Kovačević, S.; Radenković, G.; Milovanović, S.; Guzijan-Dilber, M. Rotation-free isogeometric analysis of an arbitrarily curved plane Bernoulli–Euler beam. Comput. Methods Appl. Mech. Eng. 2018, 334, 238–267. [Google Scholar] [CrossRef]
- Valizadeh, N.; Bui, T.Q.; Vu, V.T.; Thai, H.T.; Nguyen, M.N. Isogeometric simulation for buckling, free and forced vibration of orthotropic plates. Int. J. Appl. Mech. 2013, 5, 1350017. [Google Scholar] [CrossRef]
- Zou, Z.; Scott, M.A.; Miao, D.; Bischoff, M.; Oesterle, B.; Dornisch, W. An isogeometric Reissner–Mindlin shell element based on Bézier dual basis functions: Overcoming locking and improved coarse mesh accuracy. Comput. Methods Appl. Mech. Eng. 2020, 370, 113283. [Google Scholar] [CrossRef]
- Zou, Z.; Hughes, T.J.R.; Scott, M.A.; Sauer, R.A.; Savitha, E.J. Galerkin formulations of isogeometric shell analysis: Alleviating locking with Greville quadratures and higher-order elements. Comput. Methods Appl. Mech. Eng. 2021, 380, 113757. [Google Scholar] [CrossRef]
- Balobanov, V.; Kiendl, J.; Khakalo, S.; Niiranen, J. Kirchhoff–Love shells within strain gradient elasticity: Weak and strong formulations and an H3-conforming isogeometric implementation. Comput. Methods Appl. Mech. Eng. 2019, 344, 837–857. [Google Scholar] [CrossRef]
- Kruse, R.; Nguyen-Thanh, N.; Wriggers, P.; De Lorenzis, L. Isogeometric frictionless contact analysis with the third medium method. Comput. Mech. 2018, 62, 1009–1021. [Google Scholar] [CrossRef]
- Bazilevs, Y.; Pigazzini, M.S.; Ellison, A.; Kim, H. A new multi-layer approach for progressive damage simulation in composite laminates based on isogeometric analysis and Kirchhoff–Love shells. Part I: Basic theory and modeling of delamination and transverse shear. Comput. Mech. 2018, 62, 563–585. [Google Scholar] [CrossRef]
- Peng, X.; Atroshchenko, E.; Kerfriden, P.; Bordas, S.P.A. Isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth. Comput. Methods Appl. Mech. Eng. 2017, 316, 151–185. [Google Scholar] [CrossRef] [Green Version]
- Shojaee, S.; Asgharzadeh, M.; Haeri, A. Crack analysis in orthotropic media using combination of isogeometric analysis and extended finite element. Int. J. Appl. Mech. 2014, 6, 1450068. [Google Scholar] [CrossRef]
- Buffa, A.; Sangalli, G.; Vázquez, R. Isogeometric methods for computational electromagnetics: B-spline and T-spline discretizations. J. Comput. Phys. 2014, 257, 1291–1320. [Google Scholar] [CrossRef] [Green Version]
- Takizawa, K.; Tezduyar, T.E.; Otoguro, Y.; Terahara, T.; Kuraishi, T.; Hattori, H. Turbocharger flow computations with the space–time isogeometric analysis (ST-IGA). Comput. Fluids 2017, 142, 15–20. [Google Scholar] [CrossRef] [Green Version]
- Wang, C.; Wu, M.C.; Xu, F.; Hsu, M.C.; Bazilevs, Y. Modeling of a hydraulic arresting gear using fluid–structure interaction and isogeometric analysis. Comput. Fluids 2017, 142, 3–14. [Google Scholar] [CrossRef] [Green Version]
- López, J.; Anitescu, C.; Rabczuk, T. Isogeometric structural shape optimization using automatic sensitivity analysis. Appl. Math. Model. 2021, 89, 1004–1024. [Google Scholar] [CrossRef]
- Li, B.; Ding, S.; Guo, S.; Su, W.; Cheng, A.; Hong, J. A novel isogeometric topology optimization framework for planar compliant mechanisms. Appl. Math. Model. 2021, 92, 931–950. [Google Scholar] [CrossRef]
- Kim, J.; Reddy, J. 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. 2015, 226, 2973–2998. [Google Scholar] [CrossRef]
- Reddy, J.; Romanoff, J.; Loya, J.A. Nonlinear finite element analysis of functionally graded circular plates with modified couple stress theory. Eur. J. Mech.-A Solids 2016, 56, 92–104. [Google Scholar] [CrossRef]
- Wang, B.; Lu, C.; Fan, C.; Zhao, M. A stable and efficient meshfree Galerkin method with consistent integration schemes for strain gradient thin beams and plates. Thin-Walled Struct. 2020, 153, 106791. [Google Scholar] [CrossRef]
- Cazzani, A.; Malagù, M.; Turco, E.; Stochino, F. Constitutive models for strongly curved beams in the frame of isogeometric analysis. Math. Mech. Solids 2016, 21, 182–209. [Google Scholar] [CrossRef]
- Norouzzadeh, A.; Ansari, R. Isogeometric vibration analysis of functionally graded nanoplates with the consideration of nonlocal and surface effects. Thin-Walled Struct. 2018, 127, 354–372. [Google Scholar] [CrossRef]
- 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. 2017, 190, 219–241. [Google Scholar] [CrossRef]
- Fan, F.; Safaei, B.; Sahmani, S. Buckling and postbuckling response of nonlocal strain gradient porous functionally graded micro/nano-plates via NURBS-based isogeometric analysis. Thin-Walled Struct. 2021, 159, 107231. [Google Scholar] [CrossRef]
- Yu, T.; Hu, H.; Zhang, J.; Bui, T.Q. Isogeometric analysis of size-dependent effects for functionally graded microbeams by a non-classical quasi-3D theory. Thin-Walled Struct. 2019, 138, 1–14. [Google Scholar] [CrossRef]
- Yin, S.; Deng, Y.; Zhang, G.; Yu, T.; Gu, S. A new isogeometric Timoshenko beam model incorporating microstructures and surface energy effects. Math. Mech. Solids 2020, 25, 2005–2022. [Google Scholar] [CrossRef]
- Yin, S.; Deng, Y.; Yu, T.; Gu, S.; Zhang, G. Isogeometric analysis for non-classical Bernoulli-Euler beam model incorporating microstructure and surface energy effects. Appl. Math. Model. 2021, 89, 470–485. [Google Scholar] [CrossRef]
- 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. 2016, 308, 182–211. [Google Scholar] [CrossRef]
- 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 Solids 2017, 61, 164–179. [Google Scholar] [CrossRef]
- 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. 2017, 316, 328–348. [Google Scholar] [CrossRef]
- Natarajan, S.; Chakraborty, S.; Thangavel, M.; Bordas, S.; Rabczuk, T. Size dependent free flexural vibration behaviour of functionally graded nano plates. Comput. Mater. Sci. 2012, 65, 74–80. [Google Scholar] [CrossRef]
- Eringen, A.C. Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci. 1972, 10, 425–435. [Google Scholar] [CrossRef]
- Khakalo, S.; Niiranen, J. Isogeometric analysis of higher-order gradient elasticity by user elements of a commercial finite element software. Comput.-Aided Des. 2017, 82, 154–169. [Google Scholar] [CrossRef]
- Greco, L.; Cuomo, M. B-Spline interpolation of Kirchhoff-Love space rods. Comput. Methods Appl. Mech. Eng. 2013, 256, 251–269. [Google Scholar] [CrossRef] [Green Version]
- 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. 2018, 339, 137–159. [Google Scholar] [CrossRef]
- Tran, L.V.; Niiranen, J. A geometrically nonlinear Euler–Bernoulli beam model within strain gradient elasticity with isogeometric analysis and lattice structure applications. Math. Mech. Complex Syst. 2020, 8, 345–371. [Google Scholar] [CrossRef]
- Niiranen, J.; Balobanov, V.; Kiendl, J.; Hosseini, S. Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro- and nano-beam models. Math. Mech. Solids 2019, 24, 312–335. [Google Scholar] [CrossRef] [Green Version]
- Yaghoubi, S.T.; 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 Solids 2018, 69, 113–123. [Google Scholar] [CrossRef] [Green Version]
- Dym, C.L.; Shames, I.H. Solid Mechanics; Springer: Berlin/Heidelberg, Germany, 1973. [Google Scholar]
- Challamel, N.; Elishakoff, I. A brief history of first-order shear-deformable beam and plate models. Mech. Res. Commun. 2019, 102, 103389. [Google Scholar] [CrossRef]
- Shaat, M. A reduced micromorphic model for multiscale materials and its applications in wave propagation. Compos. Struct. 2018, 201, 446–454. [Google Scholar]
- Reddy, J.N. Energy Principles and Variational Methods in Applied Mechanics; John Wiley & Sons: New York, NY, USA, 2017. [Google Scholar]
- Gao, X.-L.; Mall, S. Variational solution for a cracked mosaic model of woven fabric composites. Int. J. Solids Struct. 2001, 38, 855–874. [Google Scholar] [CrossRef]
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Yin, S.; Xiao, Z.; Liu, J.; Xia, Z.; Gu, S. Variational Formulations and Isogeometric Analysis of Timoshenko–Ehrenfest Microbeam Using a Reformulated Strain Gradient Elasticity Theory. Crystals 2022, 12, 752. https://doi.org/10.3390/cryst12060752
Yin S, Xiao Z, Liu J, Xia Z, Gu S. Variational Formulations and Isogeometric Analysis of Timoshenko–Ehrenfest Microbeam Using a Reformulated Strain Gradient Elasticity Theory. Crystals. 2022; 12(6):752. https://doi.org/10.3390/cryst12060752
Chicago/Turabian StyleYin, Shuohui, Zhibing Xiao, Jingang Liu, Zixu Xia, and Shuitao Gu. 2022. "Variational Formulations and Isogeometric Analysis of Timoshenko–Ehrenfest Microbeam Using a Reformulated Strain Gradient Elasticity Theory" Crystals 12, no. 6: 752. https://doi.org/10.3390/cryst12060752