References27

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Exact solutions for elastic response in micro- and nano-beams considering strain gradient elasticity:

This paper presents the exact solutions derived for the static bending response of simply supported isotropic micro- and nano-beams. The governing differential equations of equilibrium and the asso...

Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro- and nano-beam models:

As a first step, variational formulations and governing equations with boundary conditions are derived for a pair of Euler–Bernoulli beam bending models following a simplified version of Mindlin’s ...

Abstract Simplified isotropic models of strain gradient elasticity are presented, based on the mutual relationship between the inherent (dual) gradient directions (i.e. the gradient direction of any strain gradient source and the lever arm direction of the promoted double stress). A class of gradient-symmetric materials featured by gradient directions obeying a reciprocity relation and by 4 independent h.o. (higher order) coefficients is envisioned, along with the sub-classes of hemi-collinear m...

A method of using the shifts of the beam resonant frequencies to determine the unknown parameters of nonlocal effects together with other effects such as surface elasticity, surface stress and residual stress is presented. The nonlocal effects are size-dependent, which only stand out when a specimen size diminishes. However, when the size of a specimen is small, other effects may also impact its mechanical properties and it is difficult to tell them apart. Unlike the static tests, the dynamic me...

A new non-classical third-order shear deformation model is developed for Reddy–Levinson beams using a variational formulation based on Hamilton’s principle. A modified couple stress theory and a surface elasticity theory are employed. The equations of motion and complete boundary conditions for the beam are obtained simultaneously. The new model contains a material length scale parameter to account for the microstructure effect and three surface elastic constants to describe the surface energy e...

A new Timoshenko beam model is developed using a modified couple stress theory and a surface elasticity theory. A variational formulation based on Hamilton’s principle is employed, which leads to the simultaneous determination of the equations of motion and complete boundary conditions for a Timoshenko beam. The new model contains a material length scale parameter accounting for the microstructure effect in the bulk of the beam and three surface elasticity constants describing the mechanical beh...

Correlation between beam on Winkler-Pasternak foundation and beam on elastic substrate medium with inclusion of microstructure and surface effects

A novel beam-elastic substrate element with inclusion of microstructure and surface energy effects is proposed in this paper. The modified couple stress theory is employed to account for the microstructure-dependent effect of the beam bulk material while Gurtin-Murdoch surface theory is used to capture the surface energy-dependent size effect. Interaction mechanism between the beam and the surrounding substrate medium is represented by the Winkler foundation model. The governing differential equ...

A new Bernoulli-Euler beam model based on a simplified strain gradient elasticity theory and its applications

Abstract A new Bernoulli–Euler beam model based on a simplified strain gradient elasticity theory is established in the current investigation. The generalized Euler–Lagrange equations and corresponding boundary conditions are naturally derived from the Hamilton’s principle. Then axial wave propagation of small scale bars, static bending of cantilever beams, buckling and free vibration of simply supported beams are analytically solved by using the simplified strain gradient beam theory. The influ...

A new Bernoulli–Euler beam model is developed using a modified couple stress theory and a surface elasticity theory. A variational formulation based on the principle of minimum total potential energy is employed, which leads to the simultaneous determination of the equilibrium equation and complete boundary conditions for a Bernoulli–Euler beam. The new model contains a material length scale parameter accounting for the microstructure effect in the bulk of the beam and three surface elasticity c...

Solution of Eshelby's inclusion problem with a bounded domain and Eshelby's tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory

A solution for Eshelby's inclusion problem of a finite homogeneous isotropic elastic body containing an inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). An extended Betti's reciprocal theorem and an extended Somigliana's identity based on the SSGET are proposed and utilized to solve the finite-domain inclusion problem. The solution for the disturbed displacement field is e...

Cited By13

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Static and dynamic modeling of functionally graded Euler–Bernoulli microbeams based on reformulated strain gradient elasticity theory using isogeometric analysis

Abstract null null Strain Gradient Theories (SGTs) are often considered to capture the intrinsic microstructural behaviors of the microbeams. In the present work, the recent Reformulated Strain Gradient Theory (RSGT), which accounts the effect of the strain gradient, couple stress (rotation gradients) and velocity gradients collectively, is extended for the static and dynamic modeling of Functionally Graded (FG) Euler–Bernoulli microbeams to account for the effect of the strain gradient, couple ...

On the Bending and Vibration Analysis of Functionally Graded Magneto-Electro-Elastic Timoshenko Microbeams

In this paper, a new magneto-electro-elastic functionally graded Timoshenko microbeam model is developed by using the variational formulation. The new model incorporates the extended modified couple stress theory in order to describe the microstructure effect. The power-law variation through the thickness direction of the two-phase microbeams is considered. By the direct application of the derived general formulation, the static bending and free vibration behavior of the newly developed function...

Abstract null null A new non-classical Bernoulli-Euler (B-E) beam model is developed using a simplified micromorphic elasticity theory. This micromorphic theory, which contains 7 independent material constants, is first proposed by simplifying the classical Eringen-Mindlin micromorphic theory for isotropic linear elastic materials, which includes 18 elastic constants. The new B-L beam model is then formulated by applying the simplified micromorphic theory and employing a variational approach bas...

Magnetically induced electric potential in first-order composite beams incorporating couple stress and its flexoelectric effects

A new model of a first-order composite beam with flexoelectric and piezomagnetic layers is developed. The new model is under a transverse magnetic field and can capture the couple stress and its flexoelectric effects. The governing equations are obtained through a variational approach. To illustrate the new model, the static bending problem is analytically solved based on a Navier’s technique. The numerical results reveal that the extension, deflection, and shear deformation of the current or co...

Isogeometric analysis of size-dependent Bernoulli–Euler beam based on a reformulated strain gradient elasticity theory

Abstract null null In this paper, we present an efficient and accurate numerical approach for static bending and free vibration analyses of microstructure-dependent Bernoulli-Euler beams. The current approach includes strain gradient, couple stress (rotation gradient) and velocity gradient effects simultaneously through a reformulated strain gradient elasticity, which applies only one material parameter for each gradient effect. Based on the Hamilton’s principle and an Isogeometric Analysis (IGA...

Abstract null null The unified gradient elasticity theory with applications to nano-mechanics of torsion is examined. The Reissner stationary variational principle is invoked to detect the differential and boundary conditions of equilibrium along with the consistent form of the constitutive laws. An efficient meshless numerical approach is established by making recourse to the Reissner variational functional wherein independent series solution of the kinematic and kinetic field variables are pro...

Bending, Buckling and Vibration Analysis of Complete Microstructure-Dependent Functionally Graded Material Microbeams

A new functionally graded non-classical Timoshenko microbeam model is developed by using a variational formulation. The new model incorporates strain gradient, couple stress (rotation gradient) and...

Nonlinear finite element analysis within strain gradient elasticity: Reissner-Mindlin plate theory versus three-dimensional theory

Abstract Nonlinear plate bending within Mindlin's strain gradient elasticity theory (SGT) is investigated by employing somewhat non-standard finite element methods. The main goal is to compare the bending results provided by the geometrically nonlinear three-dimensional (3D) theory and the geometrically nonlinear Reissner–Mindlin plate theory, i.e., the first-order shear deformation plate theory (FSDT), within the SGT. For the 3D theory, the nonlinear Green–Lagrange strain relations are adopted,...

A non-classical theory of elastic dielectrics incorporating couple stress and quadrupole effects: part I – reconsideration of curvature-based flexoelectricity theory:

A new non-classical theory of elastic dielectrics is developed using the couple stress and electric field gradient theories that incorporates the couple stress, quadrupole and curvature-based flexo...

A new model for producing band gaps for flexural elastic wave propagation in a periodic microbeam structure is developed using an extended transfer matrix method and a non-classical Bernoulli–Euler beam model that incorporates the strain gradient, couple stress and velocity gradient effects. The band gaps predicted by the new model depend on the three microstructure-dependent material parameters of each constituent material, the beam thickness, the unit cell length and the volume fraction. A par...