Journal ArticleDOI
The small length scale effect for a non-local cantilever beam: a paradox solved.
Noël Challamel,Chien Ming Wang +1 more
TLDR
This paper presents some simplified non-local elastic beam models, for the bending analyses of small scale rods, and shows that this paradox may be overcome with a gradient elastic model as well as an integral non-Local elastic model that is based on combining the local and the non- local curvatures in the constitutive elastic relation.Abstract:
Non-local continuum mechanics allows one to account for the small length scale effect that becomes significant when dealing with microstructures or nanostructures. This paper presents some simplified non-local elastic beam models, for the bending analyses of small scale rods. Integral-type or gradient non-local models abandon the classical assumption of locality, and admit that stress depends not only on the strain value at that point but also on the strain values of all points on the body. There is a paradox still unresolved at this stage: some bending solutions of integral-based non-local elastic beams have been found to be identical to the classical (local) solution, i.e. the small scale effect is not present at all. One example is the Euler-Bernoulli cantilever nanobeam model with a point load which has application in microelectromechanical systems and nanoelectromechanical systems as an actuator. In this paper, it will be shown that this paradox may be overcome with a gradient elastic model as well as an integral non-local elastic model that is based on combining the local and the non-local curvatures in the constitutive elastic relation. The latter model comprises the classical gradient model and Eringen's integral model, and its application produces small length scale terms in the non-local elastic cantilever beam solution.read more
Citations
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Journal ArticleDOI
A microstructure-dependent Timoshenko beam model based on a modified couple stress theory
H. M. Ma,Xin-Lin Gao,J. N. Reddy +2 more
TL;DR: In this paper, a microstructure-dependent Timoshenko beam model is developed using a variational formulation, which is based on a modified couple stress theory and Hamilton's principle.
Journal ArticleDOI
Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams
Bekir Akgöz,Ömer Civalek +1 more
TL;DR: In this article, the stability problem of nano-sized beam based on the strain gradient elasticity and couple stress theories is addressed, and the size effect on the critical buckling load is investigated.
Journal ArticleDOI
Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams
TL;DR: In this paper, it is shown that the existence of a solution of nonlocal beam elastostatic problems is an exception, the rule being non-existence for problems of applicative interest.
Journal ArticleDOI
Free vibration analysis of nonlocal strain gradient beams made of functionally graded material
Li Li,Xiaobai Li,Yujin Hu +2 more
TL;DR: In this article, a size-dependent Timoshenko beam model, which accounts for through-thickness power-law variation of a two-constituent functionally graded (FG) material, is derived in the framework of the nonlocal strain gradient theory.
Journal ArticleDOI
Bending of Euler–Bernoulli beams using Eringen’s integral formulation: A paradox resolved
TL;DR: In this article, the problem of static bending of Euler-Bernoulli beams using the Eringen integral constitutive equation is formulated, and a general method to solve the problem rigorously in integral form is proposed.
References
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Journal ArticleDOI
On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves
TL;DR: In this article, the integropartial differential equations of the linear theory of nonlocal elasticity are reduced to singular partial differential equations for a special class of physically admissible kernels.
Journal ArticleDOI
Experiments and theory in strain gradient elasticity
TL;DR: In this paper, a new set of higher-order metrics is developed to characterize strain gradient behaviors in small-scale structures and a strain gradient elastic bending theory for plane-strain beams is developed.
Journal ArticleDOI
On nonlocal elasticity
TL;DR: In this article, a theory of non-local elasticity is presented via the vehicles of global balance laws and the second law of thermodynamics via the use of a localized Clausius-Duhem inequality and a variational statement of Gibbsian global thermodynamics.
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Nonlocal polar elastic continua
TL;DR: In this article, a continuum theory of non-local polar bodies is developed for nonlinear micromorphic elastic solids, and the balance laws and jump conditions are given.
Journal ArticleDOI
Nonlocal theories for bending, buckling and vibration of beams
J. N. Reddy,J. N. Reddy +1 more
TL;DR: In this article, the Euler-Bernoulli, Timoshenko, Reddy, and Levinson beam theories are reformulated using the nonlocal differential constitutive relations of Eringen.