On a three-dimensional Kelvin problem for an elastic nonlocal medium

Summary Making use of the Eringen-Kroener form of the nonlocal constitutive equations and the exponential Fourier transformation, a system of two coupled differential equations of the second order describing the equilibrium of the body is derived. By appeal to the Helmholtz representation, the system is reduced to a single differential equation of the fourth order for the Love function, reminding a Bessel type transform of the biharmonic equation. A solution of this equation is found, and inverse transforms of the stress components using the convolution theorem established. A recourse to the formula of de la Vallée Poussin shows that, in contrast to the classical result, the stress singularity at the point of application of a concentrated force fails to appear, though the stress concentration at that point is extremely high.     

On the three-dimensional Boussinesq problem for an elastic nonlocal medium

After determining the nonlocal elastic moduli and the constitutive equations used, a brief review of the Kelvin problem in nonlocal setting is given. The Westergaard procedure of transition from the classical Kelvin problem to the classical Boussinesq problem is discussed, and applied to the nonlocal case using Fourier's exponential transformation. An example illustrating the application of the method to calculate the stress system in a nonlocal half-space is given.     

Dislocations and internal loading in a semi-infinite elastic medium with surface stresses

This paper presents analytical solutions for shear and opening dislocations in an elastic half-plane with surface stresses by using the Gurtin–Murdoch continuum theory of elastic material surfaces. The fundamental solutions corresponding to buried vertical and horizontal loads are also presented. Fourier integral transforms are used in the analysis. It is found that a characteristic length parameter that depends on the surface and bulk elastic moduli exists for this class of problems, and it represents the influence of surface stresses on the bulk elastic field. Selected numerical results are presented to demonstrate the influence of surface stresses on the bulk stress field. The fundamental solutions presented in this study can be used to develop boundary integral equation and other methods to analyze complicated fracture and boundary-value problems associated with nano-scale structures and soft elastic solids.     

Exact solution for nonlocal vibration of double-orthotropic nanoplates embedded in elastic medium

An analytical approach for free vibration analysis of all edges simply-supported double-orthotropic nanoplates is presented. The two nanoplates are assumed to be bonded by an internal elastic medium and surrounded by external elastic foundation. The governing equations are derived based on the nonlocal theory and the expressions of the natural frequencies are proposed in an explicit form. The suggested model is justified by a good agreement between the results given by present model and available data in literature. The model is used to study the vibration of double-orthotropic nanoplates for three typical deformation modes. The influences of small scale coefficient, stiffness of the external and internal mediums and aspect ratio on the frequencies of the double-orthotropic nanoplates are also elucidated.     

Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory

A single-elastic beam model has been developed to analyze the thermal vibration of single-walled carbon nanotubes (SWCNT) based on thermal elasticity mechanics, and nonlocal elasticity theory. The nonlocal elasticity takes into account the effect of small size into the formulation. Further, the SWCNT is assumed to be embedded in an elastic medium. A Winkler-type elastic foundation is employed to model the interaction of the SWCNT and the surrounding elastic medium. Differential quadrature method is being utilized and numerical solutions for thermal-vibration response of SWCNT is obtained. Influence of nonlocal small scale effects, temperature change, Winkler constant and vibration modes of the CNT on the frequency are investigated. The present study shows that for low temperature changes, the difference between local frequency and nonlocal frequency is comparatively high. With embedded CNT, for soft elastic medium and larger scale coefficients (e₀a) the nonlocal frequencies are comparatively lower. The nonlocal model-frequencies are always found smaller than the local model-frequencies at all temperature changes considered.     

On the surface waves in an elastic micropolar and microstretch medium with nonlocal cohesion

Summary A brief review of the main points of Eringen's theory of micromorphic bodies is first given, and balance equations for the linear isotropic micropolar and microstretch body are established. By appeal to the Fourier exponential transformation, nonlocal constitutive equations are derived, and assumptions with regard to the nonlocal moduli are made. The general field equations governing the propagation of a nonlocal surface wave are particularized so as to coincide with the results obtained directly in references [12], [17], [22], and [23], respectively. As an illustrative example, propagation of a microrotation and microstretch wave in a nonlocal medium in the entire Brillouin zone is examined.     

Buckling of a compressible elastic ring

Summary We study stability of a circular ring subjected to uniform hydrostatic pressure. The constitutive equations of a ring are taken in a generalized form, allowing the compression of the ring axis. We show that the null space of the linearized differential equations has dimension three. On the basis of a single bifurcation equation the bifurcation diagram is obtained. It is shown that bifurcation could be both super- and subcritical, depending on the values of parameters.     

A staggered nonlocal multiscale model for a heterogeneous medium

A staggered nonlocal multiscale model for a heterogeneous medium is developed and validated. The model is termed as staggered nonlocal in the sense that it employs current information for the point under consideration and past information from its local neighborhood. For heterogeneous materials, the concept of phase nonlocality is introduced by which nonlocal phase eigenstrains are computed using different nonlocal phase kernels. The staggered nonlocal multiscale model has been found to be insensitive to finite element mesh size and load increment size. Furthermore, the computational overhead in dealing with nonlocal information is mitigated by superior convergence of the Newton method. Copyright © 2012 John Wiley & Sons, Ltd.     

Concentrated force acting on an elastic inclusion in a thick-walled tube

A two-dimensional problem is investigated on the action of a concentrated force applied to the axis of a circular cylindrical, elastic inclusion embedded in an elastic thick-walled tube. This is a generalization of an indentation problem in infinite space, previously studied by Noble and Hussain [4] and revised by Omar and Hassan [5]. The problem is solved using a fast numerical approximation technique and numerical results are presented that allow us to evaluate the angle of contact and to establish a comparison with the case of embedding in an infinite space.     

Impact of nonlocal thin elastic plates by a cylindrical projectile

This paper is concerned with the analysis of the problem of dynamic impact of an elastic thin plate by a cylindrical projectile. The behavior of the plate material is assumed to be nonlocal elastic, and the effect of the impact is represented by a uniform velocity distribution over a circular region of the plate surface. Assuming the plate is thin, only the contributions of vertical shearing stress are considered, and the expressions of shear stress, axial displacement and the corresponding velocity components are obtained. Finally, the value of the total strain energy—crack initiation energy—for which the plastic flow will start, has been calculated and compared with experiments.     

Force on a defect in non-linear elastic medium

A derivation is given for the force per unit surface of a defect (e.g. a dislocation) in a non-linear elastic medium. The derivation is based on a Lagrangian of the same form as in linear elasticity, introduced in [1]. Principle of energy conservation is deduced on the basis of a particular case of Noether's theorem. Some peculiar properties of the Lagrangian are discussed. In the particular case of a constant Burgers vector the force reduces to the Peach-Koehler force per unit dislocation line, known in the linear theory of elasticity.     

Cohesive crack propagation in a random elastic medium

The issue of generating non-Gaussian, multivariate and correlated random fields, while preserving the internal auto-correlation structure of each single-parameter field, is discussed with reference to the problem of cohesive crack propagation. Three different fields are introduced to model the spatial variability of the Young modulus, the tensile strength of the material, and the fracture energy, respectively. Within a finite-element context, the crack-propagation phenomenon is analyzed by coupling a Monte Carlo simulation scheme with an iterative solution algorithm based on a truly-mixed variational formulation which is derived from the Hellinger–Reissner principle. The selected approach presents the advantage of exploiting the finite-element technology without the need to introduce additional modes to model the displacement discontinuity along the crack boundaries. Furthermore, the accuracy of the stress estimate pursued by the truly-mixed approach is highly desirable, the direction of crack propagation being determined on the basis of the principal-stress criterion. The numerical example of a plain concrete beam with initial crack under a three-point bending test is considered. The statistics of the response is analyzed in terms of peak load and load–mid-deflection curves, in order to investigate the effects of the uncertainties on both the carrying capacity and the post-peak behaviour. A sensitivity analysis is preliminarily performed and its results emphasize the negative effects of not accounting for the auto-correlation structure of each random field. A probabilistic method is then applied to enforce the auto-correlation without significantly altering the target marginal distributions. The novelty of the proposed approach with respect to other methods found in the literature consists of not requiring the a priori knowledge of the global correlation structure of the multivariate random field.     

On a nonlocal theory of longitudinal waves in an elastic circular bar

After determining the values of the nonlocal moduli for longitudinal waves in an infinite space, Fourier transforms of the equations of axially symmetric longitudinal waves in an infinite circularly cylindrical rod are established and decoupled according to the Pochhammer procedure. Dispersion equation is obtained from the conditions of traction free surface of the rod, and compared with its classical counterpart. While the velocity of long waves coincides, as required, with that derived in the classical case, the velocity of short waves turns out to be about 36% less.Notation a interatomic spacing - a ₁,a ₂,a ₃ coefficients defined by (2.12.5) - c wave phase velocity - d rod diameter - h, l defined by (2.15) - k wave number - overbar denotes Fourier transform - R rod radius - r, r vector of the point of observation and of generic point, respectively - u displacement in thex ₁-direction - u, w displacements in ther- andz-direction, respectively - double Fourier transform ofu(r, z, t) - (r–r) Dirac delta function - , Lamé constants - , nonlocal moduli - Fourier transforms of 03BC; and - A wave length - mass density - ₁₁ normal stress in thex ₁-direction - _rr , _rz , _zz stress components in polar coordinates,r, ,z - dilatation - , ^* defined by (2.20.2) - wave frequency - notation defined by (2.13.2) With 1 FigurePrepared with partial support of the University of Delaware.     

Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method

In this paper, the small scale effect on the vibration analysis of orthotropic single layered graphene sheets embedded in elastic medium is studied. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. Both Winkler-type and Pasternak-type foundation models are employed to simulate the interaction between the graphene sheet and surrounding elastic medium. Using the principle of virtual work the governing differential equations are derived. Differential quadrature method is employed to solve the governing differential equations for various boundary conditions. Nonlocal theories are employed to bring out the small scale effect of the nonlocal parameter on the natural frequencies of the orthotropic graphene sheets embedded in elastic medium. Further, effects of (i) nonlocal parameter, (ii) size of the graphene sheets, (iii) stiffness of surrounding elastic medium and (iv) boundary conditions on non-dimensional vibration frequencies are investigated.     

On a certain wave motion in the spatially nonlocal and nonlinear medium

The aim of this paper is to prove the existence of a finite amplitude circularly polarized plane progressive wave in the complex, nonlocal, nonlinear, homogeneous, transversely isotropic and elastic continuum consisting of two strictly interpenetrating ionic continua (in Tiersten's sense [1]). We show that if the Lennard-Jones 6–12 potential is taken into account, then as the deformation increases (the nonlinear effects), a rise in the square of the frequency occurs because of the repulsive part of the interactions.     

Propagation of elastic waves through a discontinuity in a medium

The work was supported by the Russian Fund of Fundamental Investigations (project code 93-05-9577).