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

Numerical Methods for Scientists and Engineers

A nonlocal beam theory for bending, buckling, and vibration of nanobeams

Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams

Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory

Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory

Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns

Ömer Civalek

Papers

Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams

Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory

Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns

Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory

A size-dependent shear deformation beam model based on the strain gradient elasticity theory