Journal
APPLIED MATHEMATICAL MODELLING
Volume 66, Issue -, Pages 527-547Publisher
ELSEVIER SCIENCE INC
DOI: 10.1016/j.apm.2018.09.027
Keywords
Nonlocal strain gradient theory; Static bending; Thickness effect; Shear effect
Funding
- National Natural Science Foundation of China [51605172]
- Natural Science Foundation of Hubei Province [2016CFB191]
- Fundamental Research Funds for the Central Universities [2015MS014]
- Graduates Innovation Fund of Huazhong University of Science and Technology [5003100034]
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A unified nonlocal strain gradient beam model with the thickness effect is developed to investigate the static bending behavior of micro/nano-scale porous beams. Size-dependent governing equations and corresponding analytical solutions for the bending of hinged-hinged beams are obtained by employing minimum total potential energy principle, the Navier solution method as well as the variational-consistent boundary conditions. For nonlocal strain gradient theory (NSGT) with thickness effect, virtual strain energy function of shear beams can contain additional nonlocal shear stress and high-order nonlocal shear stress related to the thickness direction in comparison with that of Euler-Bernoulli beam, so the coupling of the shear and thickness effects should be drawn huge attention. By means of detailed numerical analysis, it is found that, the stiffness-hardening effect is underestimated in NSGT without the thickness effect, and the stiffness-hardening and stiffness-softening effects of NSGT with the thickness effect can be not only length-dependent but also thickness-dependent. Interestingly, the generalized Young's modulus depends on half-wave number, which means that the generalized Young's modulus may be different due to applied load types. In the context of NSGT with the thickness effect, the deflection of Euler-Bernoulli beam predicted is smaller than that of shear beam, especially for thick beams. Furthermore, porosities distributed in the top or bottom of beams can possess a greater influence on the decrease of overall stiffness of beam than those distributed in the vicinity of the middle plane of beams. (C) 2018 Elsevier Inc. All rights reserved.
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