Effect Of Surface Elasticity Research Articles

Effect of surface elasticity on transient elastic field around a mode-III crack-tip under impact loads

We analyze the transient response of a linearly elastic solid weakened by a nanoscale crack (nanocrack) when the solid is subjected to anti-plane impact loading. We account for the nanoscale by incorporating surface elasticity into the model of deformation. The problem is formulated as a nonclassical mixed initial–boundary value problem. Most significant is the fact that both surface stress and surface mass inertia are included in the boundary condition on the crack surface. By applying Laplace and Fourier transform methods, we reduce the associated boundary value problem in the Laplace transform domain to a singular integro-differential equation. The latter is solved numerically leading to the determination of the corresponding stress intensity factors in the (Laplace) transformed domain. The Gaver–Stehfest algorithm is then applied to perform a numerical inversion of the Laplace transform from which we obtain dynamic stress intensity factors in the time domain. Numerical results characterizing the transient response of the crack-tip field are presented with particular emphasis on the influence of surface elasticity on the dynamic fracture parameters.

Surface effect on forced vibration of DNS by viscoelastic layer under a moving load

The surface effect for a forced vibration of a double-nanobeam-system (DNS) coupled by a viscoelastic layer under a moving constant load is studied in this paper. The viscoelastic layer that couples the nanobeams to each other, is modelled as spring-damper system. The Euler- Bernoulli theory and a simply supported boundary condition are considered for both nanobeams. By using the analytical solution, the dynamic displacement is obtained by considering the surface elasticity and residual tension effect on each nanobeams. Furthermore, the several significant parameters such as the velocity of the moving load, spring constant, damping coefficient and also the surface effect have been studied using some plots and examples. Finally, by observing the diagrams it was concluded that as the length of the beams reduces, the surface effect has a considerable effect on each of nanobeams especially at Nano scale, where it was not achieved by classic theories.

Effect of Curvature-Dependent Surface Elasticity on the Flexural Properties of Nanowire

Surface elasticity and residual stress strongly influence the flexural properties of nanowire due to the excessively large ratio of surface area to volume. In this work, we adopt linearized surface elasticity theory, which was proposed by Chhapadia et al., to capture the influence of surface curvature on the flexural rigidity of nanowire with rectangular cross section. Additionally, we have tried to study the bending deformation of circular nanowire. All stresses and strains are measured relative to the relaxed state in which the difference in surface residual stress between the upper and lower faces of rectangular nanowire with no external load induces additional bending. The bending curvature of nanowire in the reference and relaxed states is obtained. We find that flexural rigidity is composed of three parts. The first term is defined by the precept of continuum mechanics, and the last two terms are defined by surface elasticity. The normalized curvature increases with the decrease in height, thereby stiffening the nanowire. We also find that not only sizes but also surface curvature induced by surface residual stress influence the bending rigidity of nanowire.

Existence of solution in the bending of thin plates with Gurtin–Murdoch surface elasticity

We consider the well-posedness of classical boundary value problems in a theory of bending of thin plates which incorporates the effects of surface elasticity via the Gurtin–Murdoch surface model. We employ the fundamental solution of the governing system of equations to develop integral-type solutions of the corresponding Dirichlet, Neumann, and Robin boundary value problems. Using the boundary integral equation method, we subsequently establish results for the existence of a solution in the appropriate function spaces.

Analytical solutions for surface stress effects on buckling and post-buckling behavior of thin symmetric porous nano-plates resting on elastic foundation

Due to the high surface-to-bulk ratio, the classical continuum theory cannot accurately describe the mechanical behavior of the nanoscale structures. In this research, based on Gurtin–Murdoch surface stress theory and Kirchhoff plate model, a novel size-dependent formulation is presented for buckling and post-buckling behavior of thin symmetric porous nano-plates embedded into an elastic substrate medium. Using Hamilton’s principle, governing differential equation as well as corresponding boundary conditions is derived for rectangular porous nano-plates. For critical buckling traction and static equilibrium path, analytical solutions are developed including three case studies: simply supported porous nano-plate under biaxial tractions and uniform transverse load, simply-clamped supported porous nano-plate subjected to axial traction and uniform transverse load, and simply supported porous nano-plate under pure shear traction. In the numerical examples, effects of residual stress, surface elasticity, material porosity, and subgrade modulus are investigated for the critical buckling traction and the static equilibrium paths curves. Findings indicate that the residual stress has a significant influence on the buckling traction value and the form of the equilibrium path curve compared to the surface elasticity. Meanwhile, the buckling traction value of the nano-plate rises by reducing the material porosity and increasing the subgrade modulus.

Coupled effect of curved surface and interface on stress state of wrinkled thin film coating at the nanoscale

AbstractIn this paper, we present the model of continuum mechanics for an analysis of the size‐dependent elastic state of the wrinkled thin film coating with the nanosized thickness. For this purpose, we formulate the two‐dimensional boundary value problem for the film‐on‐substrate system under plane strain conditions in terms of the complex variable. To analyse the effect of surface and interface elasticity, the surface and interphase domains are modelled as negligibly thin layers adhering to the bulk phases without slipping. The algorithm of deriving the expressions for the stress tensor components in the first‐order approximation of the boundary perturbation method is described. Based on this algorithm, the influence of the size of the wrinkles, film‐to‐substrate stiffness ratio and film thickness on the stress concentration in the film is examined.

Investigations on the performance of piezoelectric-flexoelectric energy harvesters

Flexoelectric electromechanical systems have received recent attention for their ability to harvest energy at the nanoscale where piezoelectric systems could not generate appreciable energy. The size-dependency of flexoelectric materials has been investigated; however, the piezoelectric effect has largely been neglected due to its low performance at the nanoscale. In this study, a piezoelectric-flexoelectric reduced-order model considering material structure, size dependency, and surface smoothness effects is developed in order to determine the performance of piezoelectric and/or flexoelectric systems at different scale levels. The size dependent effects are accounted using the classical couple stress theory with surface elasticity effects modeled using the Gurtin-Murdoch theory. The surface integrity modeling is implemented to include non-smooth surfaces due to the manufacturing precision at the nano-scale. A multi-phase model is introduced to model the material structure of the nanocrystalline substrate incorporating porosity effects. At nanoscale, the combined piezoelectric and flexoelectric configuration and the flexoelectric only configuration perform similarly with little variation; however, at transition scales, from nano to micro, the combined system outperforms either piezoelectric only or flexoelectric only configuration. A non-smooth surface increases the levels of the harvested power of the system in all cases. This study shows not only the performance of flexoelectric configurations at the nanoscale, but that a piezoelectric-flexoelectric system can generate more power than previously thought near the microscale.

Semi-analytical solutions for static and dynamic responses of bi-directional functionally graded nonuniform nanobeams with surface energy effect

In this paper, the bending, buckling, and vibration behaviors of bi-directional functionally graded (BDFG) nonuniform micro/nanobeams are investigated. A new Euler–Bernoulli beam model is developed for BDFG tapered micro/nanobeams using Gurtin–Murdoch surface elasticity theory and modified couple stress theory to capture the effects of surface energy and microstructure stiffening, respectively. The present formulation accounts for the physical neutral surface. The material properties of the bulk and surface continuums of the nanobeam are assumed to vary along the thickness and length directions according to power law. Also, the cross section is assumed to vary linearly along the length direction. Hamilton principle is employed to derive the nonclassical equations of motions and boundary conditions. The generalized differential quadrature method (GDQM) is employed to accurately evaluate the variable coefficients of the obtained governing equations. Then after, the Navier’s method is employed for the simply supported BDFG nanobeam for its static bending deflection, critical buckling load, and fundamental frequency. The proposed model is validated by comparing the obtained results with available literature. Effects of different geometrical and material parameters on static and dynamic behaviors of small-scale BDFG nanobeams with the simultaneous effects of microstructure and surface elasticity are comprehensively studied and discussed. Results disclose that the nonuniformity parameters, aspect ratio, dimensionless material length-scale parameter, surface stress, surface elasticity, and gradient indices have a significant effect on the bending, buckling, and free vibration responses of BDFG tapered micro/ nanobeams.

Bounds to the pull-in voltage of a MEMS/NEMS beam with surface elasticity

The problem of pull-in instability of a cantilever micro- or nano-switch under electrostatic forces has attracted considerable attention in the literature, given its importance in designing micro- and nano-electromechanical systems (MEMS and NEMS). The non-linear nature of the problem supports the typical approach that relies on numerical or semi-analytical tools to approximate the solution.By contrast, we determine fully analytical upper and lower bounds to the pull-in instability phenomenon for a cantilever beam under the action of electrostatic, van der Waals or Casimir forces. In particular, the novel contribution of this works consists in accounting for size effects analytically, in the spirit of surface elasticity, which adds considerable complication to the problem, allowing for a nonconvex beam deflection. Surface energy effects are generally ignored in classical elasticity, although they are known to become relevant for very small structures and especially at the nano-scale, owing to their large surface/volume ratio. Closed form lower and upper bounds are given for the pull-in characteristics, that allow to discuss the role of several tuneable parameters. Indeed, the evolution of the cantilever tip deflection is presented as a function of the applied voltage up to the occurrence of pull-in and the contribution of van der Waals and Casimir intermolecular interactions is discussed. It is found that intermolecular forces strongly decrease the pull-in voltage, while surface elasticity works in the opposite direction and stabilizes the system. The accuracy of the bounding solutions, measured in terms of the difference between upper and lower analytical bounds, is generally very good, although it rapidly deteriorates as the effect of surface elasticity becomes more pronounced. Finally, approximated closed-form relations are developed to yield simple and accurate design formulae: in particular, they provide estimates for the minimum theoretical gap and for the maximum operable length for a free-standing cantilever in the presence of the effects of surface elasticity and intermolecular interactions. Results may be especially useful for designing and optimizing NEMS switches.

Large-amplitude oscillations of composite conical nanoshells with in-plane heterogeneity including surface stress effect

In the current investigation, a new formulation for nonlinear vibration behaviors of functionally graded (FG) composite conical nanoshells are constructed using Gurtin-Murdoch elasticity theory based on higher-order shear deformation shell theory (HSDFST) framework. Both Voigt and Reuss homogenization procedures are considered for the estimation of the mechanical characteristics of FG materials. Using generalized differential quadrature method (GDQM) together with Galerkin technique, the surface elastic-based nonlinear frequency-responses of FG composite conical nanoshell are obtained. It has been illustrated that the decrease of material property gradient index or transformation of boundary condition from full simply supported to full clamped, surface stress effect on the nonlinear frequency of a FG composite conical nanoshell reduces. Also, decreasing semi-vertex angle increases the frequency ratio of ωNL/ωL which reveals higher geometrical nonlinearity. However, it is seen that surface elasticity effect on the nonlinear vibration behavior of FG composite conical nanoshells is not significant.

Nonlinear Primary Frequency Response Analysis of Self-Sustaining Nanobeam Considering Surface Elasticity

This paper is focused to investigate the effects of nonlinear sources, including viscoelastic foundation and geometrical nonlinearity along with the surface elasticity and residual surface stress effects on the primary frequency response of a harmonically excited nanoscale Bernoulli-Euler beam. Due to large surface-area-to-volume ratio, the theory of surface elasticity as well as residual surface stress effects are taken into account within the beam models. The Galerkin approach accompanied by trigonometric shape functions is utilized to reduce the governing PDEs of the system to ODEs. The multiple scales perturbation method theory is applied to compute the nonlinear frequency response of nanobeam. The effects of linear and nonlinear viscoelastic damping coefficient of the medium, crystallographic directions of [100] and [111] of anodic alumina, geometrical nonlinear term and geometrical property on the nonlinear primary frequency response of nanoscale beam are investigated. The results show that theses parameters have a significant effect on the nonlinear frequency response of nanobeams in the case of primary resonance.

Visco-elastic cosmology for a sparkling universe?

We show the analogy between a generalization of the Rayleigh-Plesset equation of bubble dynamics including surface tension, elasticity and viscosity effects with a reformulation of the Friedmann-Lema\^itre set of equations describing the expansion of space in cosmology assuming a homogeneous and isotropic universe. By comparing both fluid and cosmic equations, we propose a bold generalization of the newly-derived cosmic equation mapping three continuum mechanics contributions. Conversely, the addition of a cosmological constant-like term in the fluid equation would lead also to a new phenomenology.

BIEM analysis of a graded nano‐cracked elastic half‐plane under time‐harmonic waves

AbstractThis work addresses the scattering of time‐harmonic waves by a finite, blunt nano‐crack in a graded isotropic bulk material with a free surface. The mechanical model is based on classical elastodynamic theory and describes the elastic, isotropic, graded half‐plane with quadratically varying material parameters along the depth coordinate. This model is extended to nano‐mechanics by using non‐classical boundary conditions and localized constitutive equations at the interface between crack and matrix material following the Gurtin‐Murdoch surface elasticity theory. The computational technique employed is based on the non‐hypersingular traction boundary integral equation method (BIEM) using a closed form Green's function for a half‐plane consisting of a functionally graded material (FGM). The dependences of the diffracted and scattered waves and of the local stress concentration fields on key problem parameters such as size and depth of the embedded crack, surface elasticity effects, type and characteristics of the incident wave, and the dynamic interaction phenomenon between the crack and the free‐surface boundary are all examined through parametric studies.

On size-dependent large-amplitude free oscillations of FGPM nanoshells incorporating vibrational mode interactions

At nanoscale, surface free energies of the atoms located on the free surfaces of structures significantly affect their mechanical characteristics. In this study, nonlinear large-amplitude free vibration response of nanoshells prepared from functionally graded porous materials (FGPM) is investigated by taking into account surface stress size effects and vibrational mode interactions. Non-classical shell model is constructed on the basis of the Gurtin–Murdoch type of the surface theory of elasticity having the capability of capturing surface stress size dependency. The accuracy of nonlinear vibration analysis is improved by incorporating the interaction of the main vibration mode and the first, third and fifth symmetric oscillation modes. Moreover, the closed-cell Gaussian-Random field scheme is put to use to extract the mechanical characteristics of FGPM nanoshell. Multiple timescales technique is then applied to achieve surface stress elastic-based nonlinear frequency of FGPM nanoshell analytically for different interactions between vibrational modes. It is revealed that by incorporating the interactions of the main vibration mode and higher symmetric oscillation modes, the behavior of the backbone curves belongs to the nonlinear free oscillation response of FGPM nanoshells changes from hardening to softening schema. It is found that when only the main vibration mode is taken into account, surface elasticity effects makes an enhancement in the significance of the hardening schema. However, by considering the interactions of higher symmetric oscillation modes, surface elasticity effects makes a reduction in the significance of the softening schema.

Effects of Flexoelectricity and Surface Elasticity on the Nonlinear Magnetoelectric Coupling in Unsymmetric Composites

Flexoelectricity is the development of an electric field under an applied strain gradient. This effect is highly size-dependent and is significant at nanoscales. Unsymmetric magnetoelectric (ME) composites undergo bending deformations under a magnetic field that results in non-zero strain gradients. Hence, it is important to give due consideration to flexoelectricity, along with direct piezoelectricity in these composites. This article presents a finite-element model that predicts the static and dynamic behavior of ME nanobeams subjected to magnetic fields. The effect of material nonlinearity of the ferromagnetic phase and flexoelectricity in the ferroelectric phase on the ME coupling are determined, and the obtained results show agreement with the literature. It is found that flexoelectricity results in substantially higher static ME coefficients. However, the resonant ME response is weakened by flexoelectricity. The effect of surface elasticity on the ME coupling coefficient is also investigated by making use of the Gurtin–Murdoch theory. The surface effects are found to improve the ME coefficient at bending resonance, while its effect is trivial at static and axial resonant conditions. Furthermore, the effect of load resistance on the ME voltage and charge coefficients is determined. The results indicate clear changes in the trend between short-circuit and open-circuit conditions.

Falling film with insoluble surfactants: effects of surface elasticity and surface viscosities

The stability of a thin falling film with both surface elasticity and surface viscosities induced by insoluble surfactants on its free surface is studied. Based on the full Navier–Stokes equations and surfactant concentration equation with corresponding boundary conditions, a weighted residual model (WRM) is derived to investigate the long-wave instability of the thin film incorporating the influence of surfactants. The Chebyshev spectral collocation method is employed to solve the linear stability of the film. The results show good agreement between the WRM and full equations. It is found that surface elasticity decreases the temporal growth rate and increases the critical Reynolds number, showing a stabilizing impact on the film. And the surface viscosity effect slightly reduces the growth rate and cutoff wavenumber while it does not alter the critical Reynolds number. Nonlinear travelling wave solutions are obtained using the WRM equations. As the surface elasticity is enhanced, the speed of travelling waves gradually approaches the corresponding linear neutral value, implying that the dispersion effect is damped; and the amplitudes of both fast waves and slow waves are suppressed by surface elasticity. Moreover, the bifurcation diagram of travelling waves is influenced by the surface viscosity, which basically promotes the speed of travelling waves with relatively large wavelengths. As the surface viscosity effect becomes stronger, for fast waves the amplitude of the humps slightly increases while that of the troughs becomes smaller for slow waves.

Effect of Microstructure and Surface Energy on the Static and Dynamic Characteristics of FG Timoshenko Nanobeam Embedded in an Elastic Medium

This paper presents an investigation of the size-dependent static and dynamic characteristics of functionally graded (FG) Timoshenko nanobeams embedded in a double-parameter elastic medium. Unlike existing Timoshenko nanobeam models, the combined effects of surface elasticity, residual surface stress, surface mass density and Poisson’s ratio, in addition to axial deformation, are incorporated in the newly developed model. Also, the continuous gradation through the thickness of all the properties of both bulk and surface materials is considered via power law. The Navier-type solution is developed for simply supported FG nanobeam in the form of infinite power series for bending, buckling and free vibration. The obtained results agree well with those available in the literature. In addition, selected numerical results are presented to explore the effects of the material length scale parameter, surface parameters, gradient index, elastic medium, and thickness on the static and dynamic responses of FG Timoshenko nanobeams.

Influence of surface effects on stress state in a body with two circular holes

The problem of two equal holes in a plane under uniform comprehensive tension is considered with accounting effects of surface elasticity. The problem is solved in the approximation of plane strain by using bipolar coordinates and series expansion. The problem may be of interest for the case of rather close holes. In this case, despite rather simple geometry, rather strong effects of the difference between surface and volumetric properties may be expected due to small distance between holes.

Nonlinear dynamics and stability of viscoelastic nanoplates considering residual surface stress and surface elasticity effects: a parametric excitation analysis

The present study mainly investigates surface effect on nonlinear dynamic instability of viscoelastic nanoplates under parametric excitation. In fact, great attention is given to the influence of residual surface stress on nonlinear dynamic behavior of the system. To achieve this goal, the governing equation of motion is derived by modeling a nanoplate embedded on a visco-Pasternak foundation and then, applying surface effect relations, nonlocal elasticity and nonlinear von Karman theories and Hamilton’s principle, respectively. Galerkin technique and multiple time scales method are also used to solve the equation. A class of nonlinear Mathieu–Hill equation is established to determine the bifurcations and the regions of nonlinear dynamic instability. The numerical results are performed, while the emphasis is placed on investigating the effect of residual surface stress, visco-Pasternak foundation coefficients, and parametric excitation. It is shown how residual surface stress leads to high values of amplitude response. Finally, stable and unstable regions in dynamic instability of viscoelastic nanoplates are addressed.

Surface Energy Effects on the Nonlinear Free Vibration of Functionally Graded Timoshenko Nanobeams Based on Modified Couple Stress Theory

An integrated nonlinear couple stress-surface energy continuum model is developed to study the nonlinear vibration characteristics of size-dependent functionally graded nanobeams for the first time. The nanobeam theory is formulated based on the Timoshenko kinematics, augmented by von Kármán’s geometric nonlinearity. The modified couple stress and Gurtin–Murdoch surface elasticity theories are incorporated to capture the long-range interaction and surface energy, respectively. Unlike existing Timoshenko nanobeam models, the effects of surface elasticity, residual surface stress, surface mass density and Poisson’s ratio, in addition to bending and axial deformations, are incorporated in the newly developed model. A power law function is used to model the material distribution through the thickness of the beam, considering the gradation of bulk and surface material parameters. A variational formulation of the nonlinear nonclassical governing equations and associated nonclassical boundary conditions is established by employing Hamilton’s principle. The generalized differential quadrature method is exploited in conjunction with either the Pseudo-arclength continuation or Runge–Kutta method to solve the problem with an exact implementation of the nonclassical boundary conditions. The formulation and solution procedure presented are verified by comparing the obtained results with available ones. Based on the parametric study, it is concluded that the nonclassical boundary conditions, material length scale parameter, residual surface stress, surface elasticity, bulk elasticity modulus, gradient index, nonlinear amplitude and thickness have important influences on the linear and nonlinear vibration responses of functionally graded Timoshenko nanobeams.