A new isogeometric Timoshenko beam model is developed using a modified couple stress theory (MCST) and a surface elasticity theory. The MCST is wildly used to capture microstructure effects that includes only one material length scale parameter, while the Gurtin–Murdoch surface elasticity theory containing three surface elasticity constants is employed to approximate the nature of surface energy effects. A new effective computational approach is presented for the current nonclassical Timoshenko beam model based on isogeometric analysis with high-order continuity basis functions of non-uniform rational B-splines, which effectively fulfills the higher continuity requirements in MCST. To validate the new approach and quantitatively illustrate both the microstructure and surface energy effects, the numerical results obtained from the developed approach for static deflection and natural frequencies of beams are compared with the analytical results available in the literature. Numerical results reveal that both the microstructure effect and surface energy effect should be considered in very thin beams, which also explains the size-dependent behavior.
LamDCCYangFChongACM, et al. Experiments and theory in strain gradient elasticity. J Mech Phys Solid2003; 51: 1477–1508.
2.
McFarlandAWColtonJS. Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J Micromech Microeng2005; 15: 1060–1067.
3.
ParkSKGaoXL. Bernoulli–Euler beam model based on a modified couple stress theory. J Micromech Microeng2006; 16: 2355–2359.
4.
ChenJYHuangYOrtizM. Fracture analysis of cellular materials: a strain gradient model. J Mech Phys Solid1998; 46: 789–828.
5.
HutchinsonJW. Plasticity at the micron scale. Int J Solid Struct2000; 37: 225–238.
MindlinRDTierstenHF. Effects of couple-stresses in linear elasticity. Arch Ration Mech Anal1962; 11: 415–448.
14.
KoiterWT. Couple stresses in the theory of elasticity, I and II. Nederl Akad Wetensch Proc Ser B1964; 67: 17–44.
15.
ThaiHTVoTPNguyenTK, et al. A review of continuum mechanics models for size-dependent analysis of beams and plates. Compos Struct2017; 177: 196–219.
16.
MaHMGaoXLReddyJN. A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J Mech Phys Solid2008; 56: 3379–3391.
17.
MaHMGaoXLReddyJN. A nonclassical Reddy-Levinson beam model based on a modified couple stress theory. Int J Multiscale Computat Eng2010; 8: 167–180.
18.
Salamat-talabMNateghiATorabiJ. Static and dynamic analysis of third-order shear deformation FG micro beam based on modified couple stress theory. Int J Mech Sci2012; 57: 63–73.
19.
GurtinMEMurdochAI. A continuum theory of elastic material surfaces. Arch Ration Mech Anal1975; 57: 291–323.
20.
GurtinMEMurdochAI. Surface stress in solids. Int J Solid Struct1978; 14: 431–440.
21.
GaoXLMahmoudFF. A new Bernoulli–Euler beam model incorporating microstructure and surface energy effects. Zeitschrift für angewandte Mathematik und Physik2014; 65: 393–404.
22.
GaoXL. A new Timoshenko beam model incorporating microstructure and surface energy effects. Acta Mechanica2015; 226: 457–474.
23.
GaoXLZhangGY. A microstructure-and surface energy-dependent third-order shear deformation beam model. Zeitschrift für angewandte Mathematik und Physik2015; 66: 1871–1894.
24.
ShanabRAAttiaMAMohamedSA. Nonlinear analysis of functionally graded nanoscale beams incorporating the surface energy and microstructure effects. Int J Mech Sci2017; 131: 908–923.
25.
AttiaMARahmanAAA. On vibrations of functionally graded viscoelastic nanobeams with surface effects. Int J Eng Sci2018; 127: 1–32.
26.
ShaatMMahmoudFFGaoXL, et al. Size-dependent bending analysis of Kirchhoff nano-plates based on a modified couple-stress theory including surface effects. Int J Mech Sci2014; 79: 31–37.
27.
WangKFWangBZhangC. Surface energy and thermal stress effect on nonlinear vibration of electrostatically actuated circular micro-/nanoplates based on modified couple stress theory. Acta Mechanica2017; 228: 129–140.
28.
GaoXLZhangGY. A non-classical Kirchhoff plate model incorporating microstructure, surface energy and foundation effects. Continuum Mech Thermodyn2016; 28: 195–213.
29.
ShaatMMohamedSA. Nonlinear-electrostatic analysis of micro-actuated beams based on couple stress and surface elasticity theories. Int J Mech Sci2014; 84: 208–217.
30.
WangKFKitamuraTWangB. Nonlinear pull-in instability and free vibration of micro/nanoscale plates with surface energy–a modified couple stress theory mode. Int J Mech Sci2015; 99: 288–296.
31.
ZhangGYGaoXLWangJZ. A non-classical model for circular Kirchhoff plates incorporating microstructure and surface energy effects. Acta Mechanica2015; 226: 4073–4085.
32.
KimJReddyJN. 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 Mechanica2015; 226: 2973–2998.
33.
ReddyJNRomanoffJLoyaJA. Nonlinear finite element analysis of functionally graded circular plates with modified couple stress theory. Eur J Mech A Solid2016; 56: 92–104.
BorkovićAKovačevićSRadenkovićG, et al. Rotation-free isogeometric analysis of an arbitrarily curved plane Bernoulli–Euler beam. Comput Meth Appl Mech Eng2018; 334: 238–267.
36.
Phung-VanPTranLVFerreiraAJM, et al. Nonlinear transient isogeometric analysis of smart piezoelectric functionally graded material plates based on generalized shear deformation theory under thermo-electro-mechanical loads. Nonlin Dyn2017; 87: 879–894.
37.
Nguyen-ThanhNZhouKZhuangX, et al. Isogeometric analysis of large-deformation thin shells using RHT-splines for multiple-patch coupling. Comput Meth Appl Mech Eng2017; 316: 1157–1178.
38.
GuoYHellerJHughesTJ, et al. Variationally consistent isogeometric analysis of trimmed thin shells at finite deformations, based on the STEP exchange format. Comput Meth Appl Mech Eng2018; 336: 39–79.
39.
NorouzzadehAAnsariRDarvizehM. Large elastic deformation of micromorphic shells. Part II. Isogeometric analysis. Math Mech Solid2019; 24: 1–26.
40.
KruseRNguyen-ThanhNWriggersP, et al. Isogeometric frictionless contact analysis with the third medium method. Computat Mech2018; 62: 1009–1021.
41.
BazilevsYPigazziniMSEllisonA, et al. 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. Computat Mech2018; 62: 563–585.
42.
PengXAtroshchenkoEKerfridenP, et al. Isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth. Comput Meth Appl Mech Eng2017; 316: 151–185.
43.
SinghSKSinghIVMishraBK, et al. A simple, efficient and accurate Bézier extraction based T-spline XIGA for crack simulations. Theor Appl Fract Mech2017; 88: 74–96.
44.
BuffaASangalliGVázquezR. Isogeometric methods for computational electromagnetics: B-spline and T-spline discretizations. J Computat Phys2014; 257: 1291–1320.
45.
TakizawaKTezduyarTEOtoguroY, et al. Turbocharger flow computations with the space–time isogeometric analysis (ST-IGA). Comput Fluid2017; 142: 15–20.
46.
WangCWuMCXuF, et al. Modeling of a hydraulic arresting gear using fluid–structure interaction and isogeometric analysis. Comput Fluid2017; 142: 3–14.
47.
XiaZWangYWangQ, et al. GPU parallel strategy for parameterized LSM-based topology optimization using isogeometric analysis. Struct Multidisc Optimiz2017; 56: 413–434.
48.
LieuQXLeeJ. A multi-resolution approach for multi-material topology optimization based on isogeometric analysis. Comput Meth Appl Mech Eng2017; 323: 272–302.
49.
NguyenNTHuiDLeeJ, et al. An efficient computational approach for size-dependent analysis of functionally graded nanoplates. Comput Meth Appl Mech Eng2015; 297: 191–218.
50.
YuTHuHZhangJ, et al. Isogeometric analysis of size-dependent effects for functionally graded microbeams by a non-classical quasi-3D theory. Thin Wall Struct2019; 138: 1–14.
51.
YuTZhangJHuH, et al. A novel size-dependent quasi-3D isogeometric beam model for two-directional FG microbeams analysis. Compos Struct2019; 21: 76–88.
52.
NguyenHXNguyenTNAbdel-WahabM, et al. A refined quasi-3D isogeometric analysis for functionally graded microplates based on the modified couple stress theory. Comput Meth Appl Mech Eng2017; 313: 904–940.
53.
ThaiSThaiHTVoTP, et al. Size-dependent behaviour of functionally graded microplates based on the modified strain gradient elasticity theory and isogeometric analysis. Comput Struct2017; 190: 219–241.
54.
BalobanovVNiiranenJ. Locking-free variational formulations and isogeometric analysis for the Timoshenko beam models of strain gradient and classical elasticity. Comput Meth Appl Mech Eng2018; 339: 137–159.
55.
LiuSYuTVan LichL, et al. Size and surface effects on mechanical behavior of thin nanoplates incorporating microstructures using isogeometric analysis. Comput Struct2019; 212: 173–187.
56.
SteigmannDJOgdenRW. Plane deformations of elastic solids with intrinsic boundary elasticity. Proc R Soc Lon Ser A Math Phys Eng Sci1997; 453: 853–877.
57.
SteigmannDJOgdenRW. Elastic surface-substrate interactions. Proc R Soc Lon Ser A Math Phys Eng Sci1999; 455: 437–474.
58.
ParkSKGaoXL. Variational formulation of a modified couple stress theory and its application to a simple shear problem. Z Angew Math Phys2008; 59: 904–917.
59.
HeLHLimCW. Surface green function for a soft elastic half-space: influence of surface stress. Int J Solids Struct2006; 43: 132–143.
60.
EllisRWSmithCW. A thin-plate analysis and experimental evaluation of couple-stress effects. Exp Mech1967; 7: 372–380.
NazemnezhadRSalimiMHashemiSH, et al. An analytical study on the nonlinear free vibration of nanoscale beams incorporating surface density effects. Compos B Eng2012; 43: 2893–2897.
