Nonlocal Elastic Materials Research Articles

Reflection phenomena of thermoelastic plane waves on a rigid surface within a size-dependent elastic media

The current investigation aims to comprehend the spread of time-harmonic thermoelastic plane waves in an isotropic and homogeneous nonlocal elastic material without bounds, within the framework of the Lord–Shulman model and Eringen’s nonlocal stress model. Our analysis reveals two sets of coupled longitudinal waves and one independent shear-type wave. The coupled longitudinal waves experience dispersion and damping over time due to the dissipative nature of the L-S model. In contrast, the vertically shear-type waves exhibit dispersion characteristics due to the presence of nonlocality in the medium. Furthermore, we observe that the shear-type wave encounters a critical frequency, while the coupled longitudinal waves may face critical frequency conditions under certain circumstances. We explore the reflection of thermoelastic waves at a rigid, thermally insulated, and isothermal boundary of a thermoelastic half-space, particularly in the case of an incident coupled longitudinal thermoelastic wave. We determine the amplitude ratios of the reflected waves to the incident wave through analytical methods. For a specific model, we generate various graphs to analyze the behavior of phase speeds, attenuation coefficients, and reflection coefficients. To assess the impact of the elastic nonlocal parameter on the variations of phase speeds, attenuation coefficients, and amplitude ratios of the reflected waves, we present graphical representations. Notably, our observations indicate that all waves are influenced by the nonlocality of the medium. Longitudinal waves exhibit sensitivity to thermal parameters, whereas the shear-type wave remains independent of thermal effects. Furthermore, the presence of elastic nonlocality results in a reduction in the classical shear-type wave speed.

Breakdown of smooth solutions in one dimensional nonlinear nonlocal elasticity

We write the equations of a nonlinear nonlocal elastic material in the system of Riemann invariants and view them as the perturbation of a simplified system under specific conditions that utilize a finite influence distance. For the perturbed system we study formation of singularities, a quality that passes on to the solutions of the original system.

Waves in nonlocal elastic material with double porosity

Waves in nonlocal elastic material with double porosity

Vibrations of nonlocal thermoelastic voids sphere with three–phase–lag model

The Erigen’s nonlocal elasticity model has been extended for free vibration analysis of generalized thermoelastic sphere with voids under three-phase-lag model. By applying traction-free thermal boundaries, the frequency relations have been considered to examine the endurance of reasonable modes in condensed form. To check free vibration analysis from frequency relations, the functional numerical iteration method has been employed by MATLAB software tools. The field functions i.e. natural frequencies, frequency shift and quality factor related to thermoelastic damping have been represented graphically for local and nonlocal elastic materials to pick up the quality of vibrations in several cases of interest.

On the Analysis of Free Vibrations of Nonlocal Elastic Sphere of FGM Type in Generalized Thermoelasticity

Free vibrations of non-homogenous (functionally graded material (FGM)) axisymmetric nonlocal thermoelastic hollow sphere has been taken into consideration for investigation. The material of nonlocal thermoelastic sphere is supposed to be graded using power law in radial direction. The solution of continued power series is employed to resolve the differential equations and to investigate the analytical solutions for field functions i.e. temperature, stress and displacement. The analytical results have been authenticated using numerical computations with computer based software like MATLAB. The numerical results have been shown graphically for the comparison of frequency shift and thermoelastic damping for local and nonlocal elastic materials. Deduction of results has been validated with the already published literature.

Infinitesimal deformations and stability of rods made of nonlocal elastic materials

Aim of the paper is the formulation of a criterion of infinitesimal stability for a class of rods made of nonlocal elastic materials. To that end, the nonlinear equilibrium equations of naturally straight, inextensible rods subject to terminal loads are written, and the constitutive equation assumed to represent the material response in rods of finite length is discussed. Then, the equations describing the infinitesimal deformations superimposed upon a finite one are deduced. The expression of the work done by the increments of the external loads associated with an infinitesimal deformation is employed to formulate, for the considered rods, the criterion of infinitesimal stability which does not require the existence of a stored-energy function. The criterion is applied to study the stability of simply supported rods subject to axial forces, of rods with one end clamped and the other one constrained to have the tangent parallel to the undeformed rod axis, subject to axial forces and twisting couples, and of annular rods formed from naturally straight rods with the addition of twist. The results show that rods made of nonlocal elastic materials exhibit a reduction in rigidity with respect to rods having the same geometry and made of usual elastic materials with the same tensile and shear moduli.

Integro-differential nonlocal theory of elasticity

The second-order integro-differential nonlocal theory of elasticity is established as an extension of the Eringen nonlocal integral model. The present research introduces an appropriate thermodynamically consistent model allowing for the higher-order strain gradient effects within the nonlocal theory of elasticity. The thermodynamic framework for third-grade nonlocal elastic materials is developed and employed to establish the Helmholtz free energy and the associated constitutive equations. Establishing the minimum total potential energy principle, the integro-differential conditions of dynamic equilibrium along with the associated classical and higher-order boundary conditions are derived and comprehensively discussed. A rigorous formulation of the third-grade nonlocal elastic Bernoulli–Euler nano-beam is also presented. A novel series solution based on the modified Chebyshev polynomials is introduced to examine the flexural response of the size-dependent beam. The proposed size-dependent beam model is demonstrated to reveal the stiffening or softening flexural behaviors, depending on the competitions of the characteristic length-scale parameters. The higher-order gradients of strain fields are illustrated to have more dominant effects on the beam stiffening.

Exact equilibrium solutions for nonlinear spatial deformations of nanorods with application to buckling under terminal force and couple

The paper is concerned with nonlinear spatial deformations of nanorods, described by exact solutions of the equilibrium equations of a rod theory based on the kinematics of Kirchhoff’s inextensible rods and the constitutive equation of Eringen’s nonlocal materials. The nonlinear equilibrium equations of nonlocal elastic rods that are prismatic in a natural state are established and their solutions are obtained for both configurations in which the axial curve has constant curvature and torsion, and general configurations in which the axial curve is described by means of elliptic functions. As an example of application of the theory, the buckling in space under terminal force and couple of nonlocal rods with one end clamped and the other one constrained to have the tangent parallel to the direction of the undeformed rod axis, is examined. For such rods, the eigencurves corresponding to bifurcation from configurations in which the axial curve is straight are found, and the exact solutions for the post-buckling configurations are presented. The results show that, along bifurcating branches on which the axial force is constant the rods, through a sequence of spatial equilibrium configurations, arrive at configurations in which the axial curve is a (planar) non-inflexional elastica. The comparison of the behavior of rods made of usual and nonlocal elastic materials shows that the presence of a nonlocal material has effects analogous to those due to a reduction in the rigidity of the rods.

Nonlocal elasticity: an approach based on fractional calculus

Fractional calculus is the mathematical subject dealing with integrals and derivatives of non-integer order. Although its age approaches that of classical calculus, its applications in mechanics are relatively recent and mainly related to fractional damping. Investigations using fractional spatial derivatives are even newer. In the present paper spatial fractional calculus is exploited to investigate a material whose nonlocal stress is defined as the fractional integral of the strain field. The developed fractional nonlocal elastic model is compared with standard integral nonlocal elasticity, which dates back to Eringen’s works. Analogies and differences are highlighted. The long tails of the power law kernel of fractional integrals make the mechanical behaviour of fractional nonlocal elastic materials peculiar. Peculiar are also the power law size effects yielded by the anomalous physical dimension of fractional operators. Furthermore we prove that the fractional nonlocal elastic medium can be seen as the continuum limit of a lattice model whose points are connected by three levels of springs with stiffness decaying with the power law of the distance between the connected points. Interestingly, interactions between bulk and surface material points are taken distinctly into account by the fractional model. Finally, the fractional differential equation in terms of the displacement function along with the proper static and kinematic boundary conditions are derived and solved implementing a suitable numerical algorithm. Applications to some example problems conclude the paper.

A dispersive wave equation using nonlocal elasticity

Nonlocal continuum mechanics allows one to account for the small length scale effect that becomes significant when dealing with micro- or nano-structures. This Note investigates a model of wave propagation in a nonlocal elastic material. We show that a dispersive wave equation is obtained from a nonlocal elastic constitutive law, based on a mixture of a local and a nonlocal strain. This model comprises both the classical gradient model and the Eringen's integral model. The dynamic properties of the model are discussed, and corroborate well some recent theoretical studies published to unify both static and dynamics gradient elasticity theories. Moreover, an excellent matching of the dispersive curve of the Born–Kármán model of lattice dynamics is obtained with such nonlocal model. To cite this article: N. Challamel et al., C. R. Mecanique 337 (2009).

Flexural wave propagation in single-walled carbon nanotubes

The paper presents the study on the flexural wave propagation in a single-walled carbon nanotube through the use of the continuum mechanics and the molecular dynamics simulation based on the Terroff-Brenner potential. The study focuses on the wave dispersion caused not only by the rotary inertia and the shear deformation in the model of a traditional Timoshenko beam, but also by the nonlocal elasticity characterizing the microstructure of carbon nanotube in a wide frequency range up to THz. For this purpose, the paper starts with the dynamic equation of a generalized Timoshenko beam made of the nonlocal elastic material, and then gives the dispersion relations of the flexural wave in the nonlocal elastic Timoshenko beam, the traditional Timoshenko beam and the Euler beam, respectively. Afterwards, it presents the molecular dynamics simulations for the flexural wave propagation in an armchair (5,5) and an armchair (10,10) single-walled carbon nanotubes for a wide range of wave numbers. The simulation results show that the Euler beam holds for describing the dispersion of flexural waves in the two single-walled carbon nanotubes only when the wave number is small. The Timoshenko beam provides a better prediction for the dispersion of flexural waves in the two single-walled carbon nanotubes when the wave number becomes a little bit large. Only the nonlocal elastic Timoshenko beam is able to predict the decrease of phase velocity when the wave number is so large that the microstructure of carbon nanotubes has a significant influence on the flexural wave dispersion.

Closed form solution for a nonlocal elastic bar in tension

A simple mechanical one-dimensional problem in the context of nonlocal (integral) elasticity is solved analytically. The nonlocal elastic material behaviour is described by the “Eringen model” whose nonlocality features all reside in the constitutive relation. This relation, of integral type, contains an attenuation function (usually assumed symmetric) aimed to capture the diffusion process of the nonlocality effects; it also exhibits a convolution format. The governing equation is a Fredholm integral equation of second kind whose analytical treatment, even for the usual choice of a symmetric kernel, is not easy to develop. In the present paper, assuming a specific shape for the attenuation function, a closed form solution in terms of strains is alternatively obtained by solving a Volterra integral equation of second kind. The latter can be easily solved with standard techniques, at least for the adopted kernel, taking also advantage from the symmetry of the solution. Such a closed form solution is an essential result to validate the effectiveness of numerical procedures aimed to solve more complex mechanical problems in the context of nonlocal elasticity.

Fundamental Solutions of Nonlocal Elastic Materials in Two-Dimensional Infinite and Semi-Infinite Problems.

In this paper, fundamental solutions are given for two-dimensional infinite and semi-infinite problems in one-dimensional nonlocal elastic material. Solutions for nonlocal stress and displacement are obtained by means of closed forms assuming nonlocal coefficient shape. Their solutions are illustrated graphically and compared with the classical solutions.