# Buckling and Free Vibrations of
## 1 Introduction
[PERSON] and co-workers [22; 23; 24] introduced the stress gradient theory from the integral nonlocal theories. They formulated that the stress state at a given point \(x\) depends not only on the local stress at that point but is the weighted average of the local stress of all points in the neighbourhood of \(x\). A nonlocal stress \(\sigma_{ij}^{NLoc}\) is defined as [24]:  
\[\sigma_{ij}^{NLoc}(x)=\int\alpha(\left|x^{\prime}-x\right|,\tau)\sigma_{ij}(x ^{\prime})dV(x^{\prime}) \tag{1}\]  
where \(\sigma_{ij}^{NLoc}\) is nonlocal stress tensor at point \(x\), \(\sigma_{ij}\) is classical (local) stress tensor at \(x^{\prime}\) within domain \(V\), \(\alpha\) is nonlocal modulus that depends on the distance \(|x^{\prime}-x|\) and on the dimensionless length scale \(\tau\) expressed as \(\tau=e_{0}\theta/L\) with \(e_{0}\theta\) nonlocal parameter, \(e_{0}\) nonlocal material constant, \(\theta\) and \(L\) internal and external characteristic lengths, respectively. For particular nonlocal problems originally studied in [24], instead of the nonlocal integral Equation (1), [PERSON] formulated an equivalent differential law as:  
\[\sigma_{ij}^{NLoc}-(e_{0}\theta)^{2}\ \
abla^{2}\sigma_{ij}^{NLoc}=\sigma_{ij} \tag{2}\]  
where \(\
abla^{2}\) is the Laplace operator. The problem of the equivalence between the original Eringen integral convolution Equation (1) and the differential form Equation (2) and associated mechanical paradoxes are discussed in the literature--see [1; 29; 30] for the detailed description. In the 1984 [PERSON] [31] formulated a simple model of strain gradient plasticity by introducing only one additional constitutive constant. Firstly, the strain gradient theory was applied to dislocation description. Next, in the 1990s, [PERSON] and co-workers [12; 25; 26; 32; 33] presented another simple model for strain gradient elasticity having one additional constitutive constant based on the strain gradient plasticity. They proposed gradient elasticity theory for finite deformations and infinitesimal deformations and extended the linear elastic constitutive relations with the Laplacian of the strains as [12]:  
\[\sigma_{ij}=C_{ijkl}(\epsilon_{kl}-(e_{0}\theta)^{2}\
abla^{2}\epsilon_{kl}) \tag{3}\]  
Recent achievements in mechanics of generalized continuum inform that deformation measures describing the generalized continuum, such as Cosserat theories, are kinematically redundant. The above original concept and the detailed discussion were first introduced in the paper by [PERSON] et al. [34] and next referenced in [35; 36; 37]. Applications of nonlocal continuum mechanics in case of carbon nanostructures such as carbon nanotubes and graphene were studied by [38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55] among others. The nonlocal vibrations and buckling of graphene were anyzed by [PERSON] [39], [PERSON] and co-workers [40; 41] and [PERSON] [42]. [PERSON] and [PERSON] [43] used nonlocal shear deformation theory to investigate the bending and vibration of plates. [PERSON] and [PERSON] [45] applied the nonlocal anisotropic elastic shell model for vibrations of nanostructures. [PERSON] and [PERSON] [47] studied the resonance of functionally graded micro/nano plates involving nonlocal stress and strain gradient elasticity. [PERSON] et al. [48] applied the first order shear deformation theory and the modified couple-stress theory to analyze the vibration of microplates. [PERSON] and [PERSON] [49] presented the nonlocal elastostatic problem and pointed out the nonlocality effect. [PERSON] et al. [51] applied the nonlocal theories for a large deformation of piezoelectric nanoplates. [PERSON] [53] analyzed the thermoelastic vibrations of nanoplates based on the nonlocal theory of elasticity. Currently, a monographic elaboration of nonlocal elasticity approaches is presented by [PERSON] et al. [56]. The problem of mechanical properties of nanostructures and nanocomposites in local and nonlocal regime was also previously considered by authors [57; 58; 59; 60; 61; 62; 63; 64; 65; 66].