Nonlocal theories for bending, buckling and vibration of beams

In recent years there have been many papers that considered the effects of material length scales in the study of mechanics of solids at micro- and/or nano-scales There are a number of approaches and, among them, one set of papers deals with Eringen's differential nonlocal model and another deals with the strain gradient theories The modified couple stress theory, which also accounts for a material length scale, is a form of a strain gradient theory The large body of literature that has come into existence in the last several years has created significant confusion among researchers about the length scales that these various theories contain The present paper has the objective of establishing the fact that the length scales present in nonlocal elasticity and strain gradient theory describe two entirely different physical characteristics of materials and structures at nanoscale By using two principle kernel functions, the paper further presents a theory with application examples which relates the classical nonlocal elasticity and strain gradient theory and it results in a higher-order nonlocal strain gradient theory In this theory, a higher-order nonlocal strain gradient elasticity system which considers higher-order stress gradients and strain gradient nonlocality is proposed It is based on the nonlocal effects of the strain field and first gradient strain field This theory intends to generalize the classical nonlocal elasticity theory by introducing a higher-order strain tensor with nonlocality into the stored energy function The theory is distinctive because the classical nonlocal stress theory does not include nonlocality of higher-order stresses while the common strain gradient theory only considers local higher-order strain gradients without nonlocal effects in a global sense By establishing the constitutive relation within the thermodynamic framework, the governing equations of equilibrium and all boundary conditions are derived via the variational approach Two additional kinds of parameters, the higher-order nonlocal parameters and the nonlocal gradient length coefficients are introduced to account for the size-dependent characteristics of nonlocal gradient materials at nanoscale To illustrate its application values, the theory is applied for wave propagation in a nonlocal strain gradient system and the new dispersion relations derived are presented through examples for wave propagating in Euler–Bernoulli and Timoshenko nanobeams The numerical results based on the new nonlocal strain gradient theory reveal some new findings with respect to lattice dynamics and wave propagation experiment that could not be matched by both the classical nonlocal stress model and the contemporary strain gradient theory Thus, this higher-order nonlocal strain gradient model provides an explanation to some observations in the classical and nonlocal stress theories as well as the strain gradient theory in these aspects

A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation

Wave propagation in carbon nanotubes (CNTs) is studied with two nonlocal continuum mechanics models: elastic Euler-Bernoulli and Timoshenko beam models [Philos. Mag. 41, 744 (1921)]. The small-scale effect on CNTs wave propagation dispersion relation is explicitly revealed for different CNTs wave numbers and diameters by theoretical analyses and numerical simulations. The asymptotic phase velocities and frequency are also derived from nonlocal continuum mechanics. The scale coefficient in nonlocal continuum mechanics is roughly estimated for CNTs from the obtained asymptotic frequency. In addition, the applicability and comparison of the two nonlocal elastic beam models to CNTs wave propagation are explored through numerical simulations. The research findings are proved effective in predicting small-scale effect on CNTs wave propagation with a qualitative validation study based on the published experimental reports in this field.

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Wave propagation in carbon nanotubes via nonlocal continuum mechanics

In the present study, a generalized nonlocal beam theory is proposed to study bending, buckling and free vibration of nanobeams. Nonlocal constitutive equations of Eringen are used in the formulations. After deriving governing equations, different beam theories including those of Euler–Bernoulli, Timoshenko, Reddy, Levinson and Aydogdu [Compos. Struct., 89 (2009) 94] are used as a special case in the present compact formulation without repeating derivation of governing equations each time. Effect of nonlocality and length of beams are investigated in detail for each considered problem. Present solutions can be used for the static and dynamic analyses of single-walled carbon nanotubes.

A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration

A Review on the Application of Nonlocal Elastic Models in Modeling of Carbon Nanotubes and Graphenes

A model, based on the theory of nonlocal continuum mechanics, on the column buckling of multiwalled carbon nanotubes is presented. The present analysis considers that each of the nested concentric tubes is an individual column and that the deflection of all the columns is coupled together through the van der Waals interactions between adjacent tubes. Based on this description, a condition is derived in terms of the parameters that describe the van der Waals forces and the small internal length scale effects. In particular, an explicit expression is derived for the critical axial strain of a double walled carbon nanotube which clearly demonstrates that small scale effects contribute significantly to the mechanical behavior of multiwalled carbon nanotubes and cannot be ignored.

L. J. Sudak

Papers

Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics

Theory of porous media with the advection term and mass exchange between phases