Abstract
Due to the conflict between equilibrium and constitutive requirements, Eringen’s strain-driven nonlocal integral model is not applicable to nanostructures of engineering interest. As an alternative, the stress-driven model has been recently developed. In this paper, for higher-order shear deformation beams, the ill-posed issue (i.e., excessive mandatory boundary conditions (BCs) cannot be met simultaneously) exists not only in strain-driven nonlocal models but also in stress-driven ones. The well-posedness of both the strain- and stress-driven two-phase nonlocal (TPN-StrainD and TPN-StressD) models is pertinently evidenced by formulating the static bending of curved beams made of functionally graded (FG) materials. The two-phase nonlocal integral constitutive relation is equivalent to a differential law equipped with two restriction conditions. By using the generalized differential quadrature method (GDQM), the coupling governing equations are solved numerically. The results show that the two-phase models can predict consistent scale-effects under different supported and loading conditions.
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The authors would like to acknowledge the support of the Priority Academic Program Development of Jiangsu Higher Education Institutions.
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Citation: 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
Project supported by the National Natural Science Foundation of China (No. 11672131)
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Zhang, P., Qing, H. On well-posedness of two-phase nonlocal integral models for higher-order refined shear deformation beams. Appl. Math. Mech.-Engl. Ed. 42, 931–950 (2021). https://doi.org/10.1007/s10483-021-2750-8
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DOI: https://doi.org/10.1007/s10483-021-2750-8
Key words
- well-posedness
- strain- and stress-driven two-phase nonlocal (TPN-StrainD and TPN-StressD) models
- refined shear deformation theory
- functionally graded (FG) curved beam
- generalized differential quadrature method (GDQM)