Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams
Introduction
This paper is devoted to a classification of different beam models, that account for both the shear effect and the small length scale terms for small scale applications (micro or nanostructures). The analysis is restricted to a plane motion for simplicity, and only elasticity is considered for the constitutive law. Each beam model is built consistently from variational arguments in order to obtain physically and variationally-based boundary conditions. Among the different beam models available in the literature, the simplest one is the Euler–Bernoulli beam model, where the rotation function is equal to the slope angle. The constitutive law expressed at the beam level is expressed from a linear relationship between the bending moment and the in-plane curvature. This model is useful especially for thin-beam structures but has shown limitations when modelling laminates, or sandwich structures for instance when shear effect along the depth of the beam may be predominant (or for small transverse shear stiffnesses). In the Timoshenko model which has a more refined kinematics, two independent fields are assumed for the kinematics, namely the deflections and the rotations, and the constitutive law now relates both the bending moment and the shear force to the curvature and the shear strain. Even if superior to the Euler–Bernoulli beam model, the introduction of independent rotations may be not sufficient, as considered by the Timoshenko model, since this First-Order-Shear-Deformation theory implicitly assumed that any cross-section will be plane before and after deformation. Furthermore, the Timoshenko model requires shear correction factors to compensate for the error due to the constant shear stress assumption along the depth of the beam.
Higher-order shear beam models have been developed to refine the simplified kinematics of the Timoshenko model by expressing the displacement field with polynomials of higher degree. In these refined theories, the transverse shear strain (and shear stress) is vanishing on the top and bottom edges of the beam. The higher-order shear beam models considered in this paper are the polynomial cubic-based model of Shi and Voyiadjis [1] (see more recently [2] or [3]) or the model of Bickford–Reddy ([4], [5]). Some other kinematics can be also considered such as the model of Touratier [6], the model of Karama et al. [7] and the model of Mechab [8]. Challamel [9] recently shown that the higher-order shear beam model of Shi and Voyiadjis [1] can be classified as a Timoshenko gradient elasticity model. This result has been generalized recently by Challamel et al. [10] for different kind of higher-order shear beam models, where equivalent classical and higher-order shear stiffness parameters have to be calculated. When considering the buckling behaviour of these higher-order shear columns, Wang et al. [11] also considered the buckling behaviour of higher-order Bickford–Reddy’s column ([4], [5]). Wang et al. [11] obtained a simple and elegant buckling solution that is also used in this paper. In fact, the buckling solution of Wang et al. [11] is valid for all cubic models (including the ones of [1], [2], [3], [4], [5]) that are all kinematically equivalent, even if not statically equivalent. This observation was developed by Shi [2] from numerical arguments (but with different results for the stress calculation), and the equivalence is confirmed in this paper from an analytical investigation.
In addition, small length terms can be introduced through gradient elasticity or nonlocal elasticity (typically for small scale structures). Elishakoff et al. [12] presented some gradient elasticity solutions and nonlocal elasticity solutions for the Euler–Bernoulli beam theory, the Timoshenko theory and the Shi and Voyadjis higher-order shear theory, with the specific stiffening effect of the additional gradient elasticity terms (or softening effect of the nonlocal elastic model). In the present paper, the variational arguments behind each gradient or nonlocal theory are detailed, and the governing differential equations are given for each model, with the variationally-based boundary conditions. A sensitive study valid for archetypal boundary conditions indicates that smaller is stiffer with the gradient elasticity model, whereas smaller is softener with the nonlocal elasticity model. The paper starts from the axial motion analysis, and then extends the analysis to the bending case, with the Euler–Bernoulli, the Timoshenko and the higher-order shear beam modelling. Gradient elasticity and nonlocal elasticity are detailed for each theory, and a combination of both scale effects will be also presented.
Section snippets
Static behaviour-Second-order effects neglected
Eringen’s nonlocal model [13] is defined for the uniaxial stress–strain relationship as:where σ is the uniaxial stress, ε the uniaxial strain and E the Young’s modulus. A characteristic length lc has been introduced in Eq. (1) to account for the so-called nonlocal effects. We shall discuss later the identification and the meaning of this characteristic length. As pointed out by Eringen [13], this differential equation clearly shows that the stress variable is a spatial
Higher-order shear theory
The same kind of gradient approach or nonlocal theories can be applied for beam models. We first will show that higher-order shear beam theories that include Euler–Bernoulli and Timoshenko theory are particular case of gradient elasticity Timoshenko theory.
The higher shear models considered in this paper are defined from a variational methodology and are then variationally consistent. The composite beam model is introduced from a single layer shear theory. The buckling of straight beams in the
Bending problem
The uniaxial nonlocal constitutive law Eq. (1) was introduced in the 1980s by Eringen [13], but the application of such a nonlocal theory to beam mechanics is more recent, and dates from 2003 with two pioneering papers of Peddieson et al. [41] and Sudak [42]. Multiplying the uniaxial nonlocal stress–strain constitutive law Eq. (1) by the ordinate y and integrating across the beam section leads to the following bending-curvature Eringen’s constitutive law:where
Buckling behaviour
The variational formulation of the buckling problem of Eringen’s nonlocal model of Timoshenko beam is first given by Adali [62] or Elishakoff
Conclusions
In this paper, the higher-order shear beam model is presented in a variationally-based and unified framework. Higher-order shear beam models can be useful, especially when modelling laminates, or sandwich structures for instance when shear effect along the depth of the plate may be predominant (or for small transverse shear stiffnesses). It is shown that higher-order shear beam models are in fact equivalent to gradient elasticity Timoshenko beams. The buckling and vibrations of simply supported
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