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This paper is devoted to the incorporation of rotary inertia and shear deformation in the study of acoustic metamaterials. An overwhelming majority of investigators resort to either Bernoulli–Euler or to the Timoshenko–Ehrenfest beam theories. Here, we demonstrate that the full version of the Timoshenko–Ehrenfest beam theory is not needed, and the truncated version is sufficient. An extensive numerical investigation is conducted to this end. [ABSTRACT FROM AUTHOR]

This paper investigates the macroscopic behaviour of an axially loaded discrete granular system from a stability perspective. The granular system comprises uniform grains that are elastically connected with some bending and shear interactions and confined by some elastic supports. This structural system can then be classified as a discrete repetitive system, a lattice elastic model or a Cosserat chain model. It is shown that this Cosserat chain model is exactly tantamount to the finite difference formulation of a shear-deformable Timoshenko column in interaction with a Winkler foundation. The buckling of the discrete column with pinned ends is first analytically investigated through the resolution of a finite difference equation. The solution is compared to a nonlocal approach derived by continualizing the discrete problem. The approximated Timoshenko nonlocal approach appears to be efficient with respect to the reference lattice problem and highlights some specific scale effects. This scale effect is related to the grain size with respect to the total length of the Cosserat chain. Finally, the paper shows the key role played by the shear interaction in the instabilities of granular structural system, especially when the bending interaction can be neglected. [ABSTRACT FROM AUTHOR]

In this paper, a generalized discrete elastica including both bending and shear elastic interactions is developed and its possible link with nonlocal beam continua is revealed. This lattice system can be viewed as the generalization of the Hencky bar-chain model, which can be retrieved in the case of infinite shear stiffness. The shear contribution in the discrete elastica is introduced by following the approach of Engesser (normal and shear forces are aligned with and perpendicular to the link axis, respectively) and that of Haringx (shear force is parallel to end section of links), both supported by physical arguments. The nonlinear analysis of the shearable-bendable discrete elastica under axial load is accomplished. Buckling and post-buckling of the lattice systems are analyzed in a geometrically exact framework. The buckling loads of both the discrete Engesser and Haringx elastica are analytically calculated, and the post-buckling behavior is numerically studied for large displacement. Nonlocal Timoshenko-type beam models, including both bending and shear stiffness, are then built from the continualization of the discrete systems. Analytical solutions for the fundamental buckling loads of the nonlocal Engesser and Haringx elastica models are given, and their first post-buckling paths are numerically computed and compared to those of the discrete Engesser and Haringx elastica. It is shown that the nonlocal Timoshenko-type beam models efficiently capture the scale effects associated with the shearable-bendable discrete elastica. [ABSTRACT FROM AUTHOR]

Abstract: This paper covers a large variety of theoretical generic beam models including some small length scale terms. Strain gradient elasticity and Eringen’s nonlocal elasticity models are applied to beam mechanics including Euler–Bernoulli, Timoshenko and higher-order shear beam models. The buckling and vibration behaviour of these generalized shear beam models is investigated for pinned–pinned boundary conditions. The variational formulation of these enriched beam models is given leading to consistent variationally-based boundary conditions. The paper first starts with the axial behaviour of gradient or nonlocal elasticity bars. The beam behaviour is then analyzed using a unified framework, where the kinematics classification is presented from a generalized gradient constitutive law. It is shown that higher-order shear beam models can be classified in a common gradient elasticity Timoshenko theory, whatever the shear strain distribution assumptions over the cross section. We show the kinematics equivalence between Bickford–Reddy higher-order shear beam model and Shi–Voyiadjis higher-order shear beam model, even if both models are statically not equivalent (from the stress calculation). This equivalence is highlighted on buckling and vibrations results. The model valid for macrostructures is generalized for micro or nanostructures using some nonlocal and gradient theories to account for small scale effects, in the axial and in the bending directions. We both use the Eringen’s based integral theory and the gradient theory to derive the buckling and vibration differential equations. These two theories can be connected using a generalized hybrid nonlocal law. Eringen’s model is compared to a stress gradient model, whereas the gradient elasticity theory is typically a strain gradient theory. The nonlocal framework is also developed in a variational consistent framework, for bending, vibrations and buckling configurations. The nonlocality is shown to be equivalent to higher-order inertia modelling for the dynamics analysis. Buckling and vibrations solutions are presented for the nonlocal higher-order beam/column models with pinned–pinned boundary conditions. We finally analyse the main characteristics of both nonlocal and gradient theories to capture the small scale effects for micro and nanostructures. Stiffening or softening effect of gradient or nonlocal elasticity models are discussed for the buckling and the vibrations analyses. [Copyright &y& Elsevier]

This paper is devoted to the analytical and numerical modeling of the lateral-torsional stability of horizontally layered composite beams. Composite beams are classified as horizontally layered beams with interlayer slip or sandwich beams with a weak shear core. The governing differential equations of the out-of-plane behavior of horizontally layered composite beams are supported by variational arguments. In the theoretical analysis, a distinction is made between the influence of the shear connection at the interface with respect to the in-plane or transversal deformations and to the out-of-plane or lateral deformations, respectively. Some engineering results are presented for a partially composite beam under pure bending moment. In the case of noncomposite in-plane action (orthotropic connection), a simple closed-form solution is derived for the lateral-torsional buckling moment, and it is shown that the exact dimensionless buckling moment depends only on two structural parameters for beams composed of two identical subelements. The results are analogous to those obtained for the in-plane buckling of partially composite or sandwich-type beams, where the buckling moment increases with the stiffness of the shear connection. Prandtl's valid solution for lateral-torsional buckling of ordinary beams is also found for composite beams in the case of noncomposite action in both the transversal and lateral directions. A generalization of Prandtl's valid solution for composite beams with partial composite action in the lateral direction and noncomposite action in the transversal direction is derived. It is shown that the lateral-torsional buckling formulas are strongly affected by the kinematics of the connected shear layer. Also, the lateral-torsional buckling of partially composite beams with both in-plane and out-of-plane slip behavior is analyzed using the Rayleigh-Ritz method. This mathematical problem leads to a system of differential equations with nonuniform coefficients. An approximated solution is derived for the isotropic connection with isotropic noncomposite actions, whereas an exact solution is presented for the orthotropic connection with noncomposite in-plane action. Finally, the Rayleigh-Ritz approach is compared with some numerical results associated with the exact resolution of the differential equations with nonuniform coefficients. The Rayleigh-Ritz approach appears to be efficient to capture the main phenomena, including the nonmonotonic dependence of the buckling load to the connection parameter. [ABSTRACT FROM AUTHOR]

In this paper, the buckling behavior of generic higher-order shear beam models is investigated in a unified framework. This paper shows that most higher-order shear beam models developed in the literature (polynomial, sinusoidal, exponential shear strain distribution assumptions over the cross section) can be classified in a common gradient elasticity Timoshenko theory, whatever the shear strain distribution assumptions over the cross section. The governing equations of the bending/buckling problem are obtained from a variational approach, leading to a generic sixth-order differential equation. Buckling solutions are presented for usual archetypal boundary conditions such as pinned-pinned, clamped-free, clamped-hinge, and clamped-clamped boundary conditions. The results are then extended to general boundary conditions based on generalized linear elastic connection law including vertical and rotational stiffness boundary conditions. Engineering analytical solutions are derived in a dimensionless format. The model valid for macrostructures is generalized for micro- or nanostructures using the nonlocal integral Eringen's model. The nonlocal framework is also developed in a variational consistent framework. Buckling solutions are finally presented for the nonlocal higher-order beam/colum models. [ABSTRACT FROM AUTHOR]

The buckling of higher-order shear plates is studied in this paper with a unified formalism. It is shown that usual higher-order shear plate models cm be classified as gradient elasticity Mindlin plate models, by augmenting the constitutive law with the shear strain gradient. These equivalences are useful for a hierarchical classification of usual plate theories comprising Kirchhoff plate theory, Mindlin plate theory and third-order shear plate theories. The same conclusions were derived by Challamel [Mech. Res. Commun. 38 (2011) 388] for higher-order shear beam models. A consistent variational presentation is derived for all generic plate theories, leading to meaningful buckling solutions. In particular, the variationally-based boundary conditions are obtained for general loading configurations. The buckling of the isotropic or orthotropic composite plates is then investigated analytically for simply supported plates under uniaxial or hydrostatic in-plane loading. An analytical buckling formula is derived that is common to all higher-order shear plate models. It is shown that cubic-based interpolation models for the displacement field are kinematically equivalent, and lead to the same buckling load results. This conclusion concerns for instance the plate models of Reddy [J. Appl. Mech. 51 (1984) 745] or the one of Shi [Int. J. Solids Struct. 44 (2007) 4299] even though these models are statically distinct (leading to different stress calculations along the cross-section). Finally, a numerical sensitivity study is made. [ABSTRACT FROM AUTHOR]

Abstract: Surface elasticity effects may be significant for small scale structures. In this paper, the effect of surface elasticity effects is investigated for the buckling of Euler–Bernoulli and Timoshenko beams. Engesser and Haringx theory of Timoshenko shear beams are studied. The surface elasticity effects are considered for small scale beam structures based on the Laplace–Young equation, which results in an equivalent distributed loading term in the beam equation. We show that these effects are explained by their non-conservative nature that can be essentially modelled as a follower tensile loading for inextensible beams. As a consequence the usual paradigm that smaller is stiffer is not necessarily found for these structural problems. The buckling small scale shear beams in presence of surface elasticity effects is studied for various boundary conditions. For clamped-free boundary conditions, we show that the buckling load is reduced compared to the one without this surface effect. This result is consistent with some recent numerical results based on surface Cauchy–Born model and with experimental results available in the literature. For other boundary conditions such as hinge–hinge and clamped–clamped boundary conditions, the results are identical to the ones already published. We explain in this paper the surprising results observed in the literature that surface elasticity effects may soften a nanostructure for some specific boundary conditions (due to the non-conservative nature of its loading application). Furthermore, self-instability is theoretically noticed for small shear beams. [Copyright &y& Elsevier]

Abstract: This paper is focused on elastic stability problems of partial composite columns: the conditions for the axial load not to introduce any pre-bending effects in composite columns; the equivalence, similarities and differences between different sandwich and partial composite beam theories with and without the effect of shear, with and without the effect of axial extensibility, and also the effect of eccentric axial load application. The basic modelling of the composite beam–column uses the Euler–Bernoulli beam theory and a linear constitutive law for the slip. In the analysis of this reference model, a variational formulation is used in order to derive relevant boundary conditions. The specific loading associated with no pre-bending effects before buckling is geometrically characterized, leading to analytical buckling loads of the partial composite column. The equivalence between the Hoff theory for sandwich beam–columns, the composite action theory for beam–columns with interlayer slip and the corresponding Bickford–Reddy theory, is shown from the stability point of view. Special loading configurations including eccentric axial load applications and axial loading only on one of the sub-elements of the composite beam–column are investigated and the similarity of the behaviour to that of imperfect ordinary beam–columns is demonstrated. The effect of axial extensibility on kinematical relationships (according to the Reissner theory), is analytically quantified and compared to the classical solution of the problem. Finally, the effect of incorporating shear in the analysis of composite members using the Timoshenko theory is evaluated. By using a variational formulation, the buckling behaviour of partial composite columns is analysed with respect to both the Engesser and the Haringx theory. A simplified uniform shear theory (assuming equal shear deformations in each sub-element) for the partial composite beam–column is first presented, and then a refined differential shear theory (assuming individual shear deformations in each sub-element) is evaluated. The paper concludes with a discussion on this shear effect, the differences between the shear theories presented and when the shear effect can be neglected. [Copyright &y& Elsevier]

Golmakani and Rezatalab [1] suggested in their paper that the deflection of a simply supported nonlocal elastic plate under uniform load is not affected by the small length scale terms. They based their proof on the use of Navier's method using a sinusoidal-based deflection solution. This insensitivity of the deflection solution of a simply supported nonlocal elastic plate with respect to the small length terms of Eringen's model is not correct, as already detailed in the literature (for example, see [2] for beam problems). In fact, the deflection of the nonlocal plate (in the Eringen sense) is larger than the one of the local case, as shown in many papers available in the literature. We prove in this reply to the authors that the Navier's method has to be correctly applied for highlighting the specific sensitivity phenomenon of the deflection solution, as compared to exact analytical solution. The problem is exactly the same as for the bending of a beam. [ABSTRACT FROM AUTHOR]

• This paper presents a brief history of beam and plate models in elasticity, that take into account the rotary inertia and shear contribution. • We show that Bresse (1859) already in the mid XIXth century rigorously derived the set of equations for curved shear and axially extensible beams in dynamics. • These equations have been generalized by Timoshenko (1913; 1916; 1920; 1921; 1922) by a slight modification of the shear stiffness, that incorporates a shear correction factor generally different from unity. • The calibration of the shear correction factor with respect to three-dimensional elasticity solutions has been elaborated jointly by Ehrenfest and Timoshenko in the second decade of the XXth century (the calibration of the shear correction factor for rectangular cross section has been finalized by Timoshenko in 1922, even if the basics results were available in his letter dated from 1913, in collaboration with Ehrenfest). • Whereas Bresse–Timoshenko–Ehrenfest equations have been elaborated mainly during more than half a century between 1859 and 1922 (even if researches on shear beam theories were still active), the Uflyand–Mindlin plate theory has been built in a more contracted period, mainly between 1944 and 1951. • Beam and plate equations are presented in a unified way, with new historical analyses and in a very concise perspective, that may be useful for scientists in the physics and mechanics community. It gives a new insight into what is commonly called the Timoshenko beam theory. This paper presents a brief history of beam and plate models in elasticity, that take into account the rotary inertia and shear contribution. We show that Bresse (1859) already in the mid XIXth century rigorously derived the set of equations for curved shear and axially extensible beams in dynamics, although without shear correction factor. When restricted to straight beams, Bresse (1859) obtained a two-field beam kinematics composed of independent deflection and rotation variables. These equations have been generalized by Timoshenko (1913; 1916; 1920; 1921; 1922), without direct reference to the works of Bresse, by a slight modification of the shear stiffness, that incorporates a shear correction factor generally different from unity. The calibration of the shear correction factor with respect to three-dimensional elasticity solutions has been elaborated jointly by Ehrenfest and Timoshenko in the second decade of the XXth century (the calibration of the shear correction factor for rectangular cross section has been finalized by Timoshenko in 1922, even if the basic results were available in his letter dated from 1913, in collaboration with Ehrenfest). Whereas Bresse-Timoshenko–Ehrenfest equations have been elaborated mainly during more than half a century between 1859 and 1922 (even if researches on shear beam theories were still active), the Uflyand–Mindlin plate theory has been built in a more compact period, mainly between 1944 and 1951. Whereas Reissner (1944, 1945) developed a static (or stress-based) formulation, the kinematic theory of Uflyand-Mindlin plate model has been first elaborated by Bolle (1947) and Hencky (1947) for static setting, before the full generalization to dynamics by Uflyand a year later, in 1948, and its complete variational derivation by Mindlin (1951). A lot of efforts were spent since almost half of the century to calibrate the shear correction factor and to enrich the kinematics of Uflyand-Mindlin plate theory. It is quite surprising that the shear correction factors of both the Bresse–Timoshenko–Ehrenfest beam model and its Uflyand-Mindlin plate analogy were already implicitly available in the paper of Timoshenko (1922), performed, in his own testimony with Ehrenfest. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of the present work it to study the damping and forced vibrations of three-layered, symmetric laminated composite beams. In the analytical formulation, both normal and shear deformations are considered in the core by using the higher-order zig-zag theories. The harmonic balance method is coupled with a one mode Galerkin procedure for a simply supported beam. The geometrically nonlinear coupling leads to a nonlinear frequency amplitude equation governed by several complex coefficients. In the first part of the paper, linear and nonlinear damping parameters of laminated composite beams are obtained. In the second part, nonlinear forced vibration analysis is carried out for small and large vibration amplitudes. The frequency response curves are presented and discussed for various geometric and material properties. [Copyright &y& Elsevier]