A new isogeometric Timoshenko beam model incorporating microstructures and surface energy effects

2020 ◽  
Vol 25 (10) ◽  
pp. 2005-2022 ◽  
Author(s):  
Shuohui Yin ◽  
Yang Deng ◽  
Gongye Zhang ◽  
Tiantang Yu ◽  
Shuitao Gu
Keyword(s):  
Surface Energy ◽  
Elasticity Theory ◽  
Timoshenko Beam ◽  
Couple Stress Theory ◽  
Surface Elasticity ◽  
Numerical Results ◽  
Beam Model ◽  
Timoshenko Beam Model ◽  
Order Continuity ◽  
Microstructure Effect

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.

scholarly journals On the Generalization of the Timoshenko Beam Model Based on the Micropolar Linear Theory: Static Case

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Andrea Nobili
Keyword(s):  
Linear Theory ◽  
Timoshenko Beam ◽  
Couple Stress ◽  
Material Parameter ◽  
Couple Stress Theory ◽  
Beam Model ◽  
Static Case ◽  
Timoshenko Beam Model ◽  
Stress Theory ◽  
Modified Couple Stress

Three generalizations of the Timoshenko beam model according to the linear theory of micropolar elasticity or its special cases, that is, the couple stress theory or the modified couple stress theory, recently developed in the literature, are investigated and compared. The analysis is carried out in a variational setting, making use of Hamilton’s principle. It is shown that both the Timoshenko and the (possibly modified) couple stress models are based on a microstructural kinematics which is governed by kinosthenic (ignorable) terms in the Lagrangian. Despite their difference, all models bring in a beam-plane theory only one microstructural material parameter. Besides, the micropolar model formally reduces to the couple stress model upon introducing the proper constraint on the microstructure kinematics, although the material parameter is generally different. Line loading on the microstructure results in a nonconservative force potential. Finally, the Hamiltonian form of the micropolar beam model is derived and the canonical equations are presented along with their general solution. The latter exhibits a general oscillatory pattern for the microstructure rotation and stress, whose behavior matches the numerical findings.

scholarly journals A non-classical Mindlin plate model incorporating microstructure, surface energy and foundation effects

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160275 ◽  
Author(s):  
X.-L. Gao ◽  
G. Y. Zhang
Keyword(s):  
Surface Energy ◽  
Classical Model ◽  
Surface Elasticity ◽  
Mindlin Plate ◽  
Beam Model ◽  
Plate Model ◽  
Energy Effect ◽  
Special Cases ◽  
Microstructure Effect ◽  
Surface Energy Effect

A non-classical model for a Mindlin plate resting on an elastic foundation is developed in a general form using a modified couple stress theory, a surface elasticity theory and a two-parameter Winkler–Pasternak foundation model. It includes all five kinematic variables possible for a Mindlin plate. The equations of motion and the complete boundary conditions are obtained simultaneously through a variational formulation based on Hamilton's principle, and the microstructure, surface energy and foundation effects are treated in a unified manner. The newly developed model contains one material length-scale parameter to describe the microstructure effect, three surface elastic constants to account for the surface energy effect, and two foundation parameters to capture the foundation effect. The current non-classical plate model reduces to its classical elasticity-based counterpart when the microstructure, surface energy and foundation effects are all suppressed. In addition, the new model includes the Mindlin plate models considering the microstructure dependence or the surface energy effect or the foundation influence alone as special cases, recovers the Kirchhoff plate model incorporating the microstructure, surface energy and foundation effects, and degenerates to the Timoshenko beam model including the microstructure effect. To illustrate the new Mindlin plate model, the static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the general formulae derived.

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