Surface Elasticity Theory Research Articles

Axisymmetric Hertzian contact problem accounting for surface tension and strain gradient elasticity

In this paper, we investigate the axisymmetric Hertzian contact problem at micro-/nanoscale. The deformation of material bulk is described by a simplified theory of strain gradient elasticity, and the influence of surface tension is integrated based on the surface elasticity theory. Using the Mindlin's potential function method and double Fourier integral transform, the normal surface displacement induced by a concentrated force is derived in a closed form. Following this, the contact between a rigid sphere and an elastic half-space is formulated in terms of singular integral equation, which is numerically solved by applying the Gauss-Chebyshev method. The results indicate that the distribution of contact pressure is distinctly different from that in classical elasticity theory. The indented substrate tends to perform stiffer due to the effects of surface tension and strain gradient elasticity. When the contact radius is comparable with the material length parameter, the indentation force (depth) can be ten (three) times of that given by classical Hertz theory.

Characteristics of wave propagation in pre-stressed viscoelastic Timoshenko nanobeams with surface stress and magnetic field influences

The current investigation explores the behavior of pre-stressed viscoelastic Timoshenko nanobeams under the influence of surface effects and a longitudinal magnetic field. Utilizing a modified version of non-local strain gradient theory through the Kelvin–Voigt viscoelastic model, a closed-form dispersion relation using a suitable analytical approach has been derived. To account for surface stresses, Gurtin–Murdouch surface elasticity theory has been employed. Additionally, the study delves into the impact of a longitudinal magnetic field on a single-walled carbon nanotube, considering Lorentz magnetic forces. The validity of the findings is established by deriving results in the absence of surface effects and magnetic fields, aligning well with existing literature. The investigation indicates that pre-stress has marginal effects on flexural and shear waves, while surface effects, magnetic fields, non-locality, characteristic length, and nanotube diameter significantly influence the phase velocity. Additionally, the threshold velocity and blocking diameter are discussed for the model.

Study on the attributes of transverse surface waves in a layered piezoelectric semi-space with surface elasticity theory and extended stroh formalism

Previous works on surface acoustic wave sensors have shown great limitations in selecting piezoelectric materials and the wave propagation direction. To eliminate such limitations in the technological revolution of SAW sensors, the current paper's main purpose is to explore how wave propagation orientation affects the performance of SAW macro- and nano-sensors. Based on Extended Stroh formalism, the theoretical forms are derived and exploited to present the wavenumber of transverse waves in an arbitrary direction of the piezoelectric medium. Furthermore, we consider surface elasticity theory to acquire the phase velocity equation on the basis of the expression of the wavenumber. More intuitively, a physical model is set up to obtain the horizontal shear stiffness of the surface and bulk layers. Then, the numerical case is carried out to determine the relationship between phase velocity and wave propagation orientation. By comparing the numerical study and the physical model, it can be found that the empirical formula of shear stiffness for bulk and surface layers offers a helpful route to precisely predict the mechanical attributes of SAW macro- and nano-sensors, respectively. The summaries of the current theoretical work benefit the manufacturing of surface acoustic wave sensors with improved performance.

Surface effect on the partial-slip contact of a nano-sized flat indenter

The partial-slip contact at nano-scale is always influenced by a surface effect. A new model based on the Gurtin-Murdoch (G-M) surface elasticity theory is proposed in this paper to study the partial-slip contact behavior of a rigid flat nano-indenter on an elastic half-space. The surface effect in the partial-slip contact is characterized via a non-classical boundary condition involving a surface-induced normal traction related to the residual surface stress and a surface-induced tangential traction related to the surface elasticity, the influences of which on stress and displacement fields and the stick-slip state at the contact surface are investigated. It is found that, with the increase of residual surface stress, both the normal pressure and vertical and horizontal displacements in the contact zone decrease, the relative slip between indenter and substrate reduces and the stick region is enlarged. Such effects are basically attributed to the action of the surface-induced normal traction opposite to the externally applied compression. However, the increase of surface elastic constants is conductive to the relative slip and results in a shrinkage of the stick region, which is attributed to the action of the surface-induced tangential traction opposite to the frictional stress. An interesting phenomenon is further unveiled that, when the frictional coefficient increases, the dominant role in affecting the stick-slip state changes from the surface-induced tangential traction to the normal one, thus inspiring a feasible route to manipulate the surface effect by tuning the frictional coefficient of substrate. The present research enables one to better understand the partial-slip contact behavior of nano-indenters, which is of guiding value for anti-wear designs in nano-mechanical devices.

A new model for spatial rods incorporating surface energy effects

A new non-classical model for spatial rods incorporating surface energy effects is developed using a surface elasticity theory. A variational formulation based on the principle of minimum total potential energy is employed, which leads to the simultaneous determination of the equilibrium equations and complete boundary conditions. The newly developed spatial rod model contains three surface elasticity constants to account for surface energy effects. The new model recovers the classical elasticity-based Kirchhoff rod model as a special case when the surface energy effects are not considered. To illustrate the new spatial rod model, two sample problems are analytically solved by directly applying the general formulas derived. The first one is the buckling of an elastic rod of circular cross-section with fixed-pinned supports, and the other is the equilibrium analysis of a helical rod deformed from a straight rod. An analytical formula is derived for the critical buckling load required to perturb the axially compressed straight rod, and two closed-form expressions are obtained for the force and couple needed in deforming the helical rod. These formulas reduce to those based on classical elasticity when the surface energy effects are suppressed. For the buckling problem, it is found that the critical buckling load predicted by the current new model is always higher than that given by the classical elasticity-based model, and the difference between the two sets of predicted values is significantly large when the radius of the rod is sufficiently small but diminishes as the rod radius increases. For the helical rod problem, the numerical results reveal that the force and couple predicted by the current model are, respectively, significantly larger and smaller than those predicted by the classical model when the rod radius is very small, but the difference is diminishing with the increase of the rod radius.

Bending edge wave on a functionally graded fluid‐saturated porous magneto‐electro‐elastic plate

AbstractThis study investigates the localized wave propagation at the free edge of a thin semi‐infinite Magneto‐Electro‐Elastic (MEE) plate. The plate is supported by a two‐parameters elastic foundation. Biot's porosity theory is considered to formulate the porosity in the model, and the pore structure is saturated with an incompressible fluid. The Kirchhoff–Love plate theory determines the plate displacement field. A non‐homogeneity in the material parameters is considered, which varies with the spatial coordinate along the thickness direction of the plate. In order to illustrate the impacts of the surface elements on bending wave propagation, the model incorporates the surface elasticity theory. The short circuit boundary conditions are implemented to derive the general form of the potential function connected with the electric and magnetic fields, respectively. The dispersion relation for bending edge wave on a MEE fluid‐saturated porous plate is obtained by considering a sinusoidal waveform, which propagates along the free edge of the plate. A numerical method named Finite Difference (FD) scheme is employed to analyze the stability of the numerical scheme and also extract the expression of the velocity profiles of the bending edge wave in the absence of the pore structure. The impacts of the various parameters included in the considered model are examined using a numerical example, from which conclusions regarding their sensitivity are derived.

Intermolecular forces and surface stress residual effects on the size-dependent performance analysis of nanocrystalline micro/nano gyroscopes

This work investigates an electrostatically-actuated vibrating beam micro/nano-gyroscope made of nanocrystalline silicon material. The comprehensive size-dependent micro/nanogyroscope model is developed by couple stress and surface elasticity theories. Then, considering the effects of intermolecular forces (Casimir force and van der Waals force) and surface stress residual, an in-depth investigation was conducted on the performance of the micro/nanogyroscope. The calculation results display that the maximum static deflection, pull-in voltage and natural frequency change obviously with the increase of surface stress residual as the scale varies from micrometer to nanoscale. However, the effect of intermolecular forces on the performance of gyroscopes is exactly the opposite, which improves the prediction of system structural softening deformation phenomenon. Considering the influence of the Casimir and van der Waals forces in micro/nano gyroscopic system will help gyroscopes to obtain reliability and functionality sensitivity, increase the accuracy of predicting static deformation, natural frequency and differential frequency. Moreover, a frequency-domain method using the difference between the two vibration natural frequencies is studied, with changing the scale of the gyroscope under the effect of the intermolecular forces. The numerical simulations exhibit that when considering smaller structure sizes of the gyroscope, strong coupling occurs between the intermolecular forces and surface stress residual, which is helpful for the design of micro/nano gyroscopes.

Dynamic BEM analysis of elastic anisotropic nano‐sheet with surface imperfections and cavities

AbstractThe paper considers elastic anisotropic nano‐sheet with nanotopography and containing embedded nanocavities under time‐harmonic in‐plane wave motion. The mechanical model is based on the classical elastodynamic theory for the anisotropic bulk solid and the nonclassical boundary conditions along its boundary derived by Gurtin and Murdoch. A localized constitutive equation for the half‐plane boundary is considered in the frame of surface elasticity theory. The computational tool is an efficient direct displacement boundary element method (BEM) based on the frequency‐dependent elastodynamic fundamental solution for elastic generally anisotropic material. To show the versatility of the proposed methodology, wave propagation in an elastic anisotropic heterogeneous nano‐sheet with nanorelief presented by hill‐canyon nanotopography is studied. The simulations illustrate the dependence of the wave field on the material anisotropy, on the surface elasticity properties, on the nanorelief peculiarities, on the nanocavities existence and their dynamic interaction, and on the incident wave characteristics. The obtained results reveal the potential of the developed mechanical model based on the boundary integral equations in the frame of surface elasticity theory to produce highly accurate results by using strongly reduced discretization mesh in comparison with the domain‐based methods.

Large Deflection and Post-Buckling Analysis of Cantilever Nanorods Including Effects of Couple and Surface Stresses by Intrinsic Coordinate Finite Elements

This paper presents a novel approach to the analysis of large deflection and post-buckling behavior of cantilever nanorods under different load conditions. The proposed approach utilizes a variational method, incorporating the Gurtin–Murdoch surface elasticity theory and the consistent couple stress theory to account for size-dependency effects. By accounting for the strain energy contributions of the bulk material, surface layer, and load conditions, this approach expresses the behavior of the nanorods in terms of their intrinsic coordinates. A finite element method was used to solve this numerical problem, generating a system of non-linear equations that were iteratively solved using the Newton–Raphson method. The results obtained from the finite element method were confirmed by those from the shooting method. Also, it highlights the effectiveness of the variational model in predicting the large deflection and post-buckling behavior of cantilever nanorods while considering both surface stress and couple stress effects. Furthermore, the study investigated the influence of surface stress and couple stress on the response of the nanorods to point and uniform loads. The results showed that incorporating both effects enhanced the stiffness of the nanorods compared to scenarios in which these effects were neglected. In addition, the couple stress effect was found to have a greater influence on the stiffening of nanorods for the same value of unitless parameters compared to surface stress effects. These findings offer valuable insights into the large deflection and post-buckling behavior of cantilever nanorods and highlight the importance of surface stress and couple stress effects. The model proposed in this study has potential applications in advanced technological device designs by providing a comprehensive understanding of the mechanical response of nanorods, allowing for more accurate predictions and enhanced device performance.

Exact solution of thermal postbuckling of porous piezoelectric nanobeams with surface effect

In this paper, the thermal postbuckling of porous piezoelectric nanobeams is analyzed using the first-order shear deformation beam theory based on the surface elasticity theory and "core-shell" model, where the surface effect is introduced by employing the surface energy model. The governing equations for thermal postbuckling of porous piezoelectric nanobeams were derived by the principle of minimum potential energy. It is found that surface effect increases the effective elastic modulus and dimensionless critical load and reduces the thermal postbuckling conformations, thermal postbuckling paths and amount of induced charge. Meanwhile, the reasonable pore distribution pattern can effectively improve the mechanical properties.

The influence of surface effects on axisymmetric vibration of graded porous Mindlin circular nanoplate in a temperature field

AbstractThe surface effect significantly affects the mechanical response of nanoscale materials. In this work, we introduce Gurtin‐Murdoch surface elasticity theory into Mindlin plate theory to study the axisymmetric vibration characteristic of graded porous circular nanoplate in a temperature field. The main structure is a porous graded material with uniform or non‐uniform porosity distribution in its thickness. The numerical shooting technique was applied to solve the governing differential equations derived from Hamilton's principle. Responses for the vibration characteristic of the graded porous nanoplate for both clamped and simply supported boundary conditions were obtained. The numerical results show that when the surface effect is removed, the classical results of Mindlin circular plate will be obtained. Natural frequencies and mode shapes varying with thermal and porosity coefficients were present graphically. And then the effects of surface material properties on the axisymmetric vibration characteristic of the nanoplates were discussed in detail. The parametric study examined the influence of porosity coefficient, porosity distribution mode, surface elastic parameters, and residual surface stress on the vibration characteristics of porous circular nanoplates.

Dynamic Analysis of a Timoshenko–Ehrenfest Single-Walled Carbon Nanotube in the Presence of Surface Effects: The Truncated Theory

The main objective of this paper is to study the free vibration of a Timoshenko–Ehrenfest single-walled carbon nanotube based on the nonlocal theory and taking surface effects into account. To model these effects on frequency response of nanotubes, we use Eringen’s nonlocal elastic theory and surface elastic theory proposed by Gurtin and Murdoch to modify the governing equation. A modified version of Timoshenko nonlocal elasticity theory—known as the nonlocal truncated Timoshenko beam theory—is put forth to investigate the free vibration behavior of single-walled carbon nanotubes (SWCNTs). Using Hamilton’s principle, the governing equations and the corresponding boundary conditions are derived. Finally, to check the accuracy and validity of the proposed method, some numerical examples are carried out. The impacts of the nonlocal coefficient, surface effects, and nanotube length on the free vibration of single-walled carbon nanotubes (SWCNTs) are evaluated, and the results are compared with those found in the literature. The findings indicate that the length of the nanotube, the nonlocal parameter, and the surface effect all play important roles and should not be disregarded in the vibrational analysis of nanotubes. Finally, the results show how effective and successful the current formulation is at explaining the behavior of nanobeams.

Analytical solutions for the plane thermoelastic problem of a nano-open crack in one-dimensional hexagonal quasicrystal non-periodic plane

As a stable material at high temperatures, the presence of cracks and other defects greatly reduces the performance of quasicrystals (QCs). The plane thermoelastic problem of one-dimensional hexagonal quasicrystals (1DHQ) non-periodic plane containing a nano-open crack is analyzed using surface elasticity theory. The closed form solutions are obtained to describe the thermoelastic field corresponding to the nano-crack by the Fourier integral transformation technique. The analytical solutions of the thermal stress intensity factors (TSIFs) and the strain energy density factor (SEDF) at the crack tip are derived explicitly. The influence of the applied loads, the surface effects and the multi-field coupling effects on the TSIFs and SEDF are discussed. The numerical results reveal that the surface effects have an obvious impact on TSIFs with small applied loads, and the surface effects will inhibit the propagation of nano-open cracks. The research results of this study provide some reliable theoretical references for the development of nano-QCs.

A comparative study of reflection and refraction of [formula omitted] waves across a tri-layered thin-walled micro-structure

The current paper uses mathematical analysis to investigate the behaviour of reflected and transmitted SH-waves through a three-level structure with interface comprises of thin-walled micro-structure. The geometry of the structure consists of a slab with finite thickness sandwiched between two infinite spaces. Using the couple stress theory (Mindlin and Tiersten in Arch. Ration. Mech. Anal. 11:415–448, 1962), the mechanical behaviour of the successive interfaces has been characterized by means of two cases namely surface elasticity model and strain-gradient thin membrane model. The closed form expression of Transmission Coefficient (TC), Reflection Coefficient (RC), and the phase angle have been derived by means of suitable analytical treatment in each of the two models. Furthermore, they are reduced to classical elasticity model to validate the finding. Several special cases have been also derived and discussed to support the current investigation. Numerical results have been simulated by categorizing incident SH-wave through rarer to denser and denser to rarer medium in all possible orders. The core objective of the paper is to illustrate the influence of interfacial elasticity parameters, phase angles, characteristic length, and wavenumbers on TC and RC for both the models. This has been supported by the help of contour plots. A region of Total Internal Reflection (TIR) has been obtained which critically refracts the transmitted waves along the interface. It should be pointed out that the present study involves multi-field knowledge of surface waves, couple stress theory, and surface elasticity theory.

Influence of surface effect on post-buckling behavior of piezoelectric nanobeams

Piezoelectric nanobeams with excellent mechanical, thermal and electrical properties are important components in micro-nano electromechanical systems, which are widely used as sensors, brakes and resonators. Based on the Euler–Bernoulli beam model, the influence of surface effect on the post-buckling behaviour of piezoelectric nanobeams is analysed. According to the surface elasticity theory and the ‘core–shell’ model, the surface energy model is used to introduce the influence of surface effect. The governing equations and boundary conditions of the post-buckling of piezoelectric nanobeams under the influence of surface effect are derived by the principle of minimum potential energy. The analytical solution of post-buckling is obtained by the eigenvalue method. The influence of surface effect on the post-buckling configuration, post-buckling path, amount of induced charge and critical load of piezoelectric nanobeams with different external constraints and cross-sectional dimensions are discussed. The results show that surface effect has a significant influence on the post-buckling of piezoelectric nanobeams. Considering surface effect, the effective elastic modulus and critical load of piezoelectric nanobeams are increased, and the post-buckling configuration, post-buckling path and amount of induced charge are reduced. These findings contribute to the study of micro-nano electromechanical systems based on nanobeam structures and provide a theoretical basis for the design and manufacture of nanodevices.

Surface Effect in Nano-Scale Fretting Contact Problems

Abstract The fretting contact behavior of nanostructured materials is significantly influenced by the surface effect. A model of fretting contact between a nano-sized rigid cylindrical indenter and an elastic half-plane is established based on Gurtin–Murdoch (G–M) surface elasticity theory, with which the surface effects on the stress and displacement distributions and the size of stick region (no-slip region) in the contact zone are systematically studied. It is found that the surface effect induces an additional traction besides the external force applied by punch, which could help to smoothen the stress and displacement distributions. The normal surface-induced traction related to the residual surface stress is opposite to the externally applied compression, which results in a material stiffening in the contact zone so that the contact radius, normal displacement, and normal stress decrease compared with their classical counterparts. The tangential surface-induced traction is also opposite to the externally applied frictional stress, consequently leading to reductions of the shear stress and tangential displacement induced by friction in the contact zone. More interestingly, the surface effect leads to three possible states in the contact zone, including complete slip, partial slip, and complete stick, instead of the solely partial slip state in classical fretting contact models without surface effect. Among them, the complete stick due to the action of surface residual stress is more beneficial for inhibiting the wear of contact devices, which can be realized by reducing the indenter size. The present research does not only help one to better understand the physical mechanism in nano-scale fretting contact problems, but should also guide the anti-wear design in nano-electro-mechanical (NEMs) systems.

Hybrid homogenization theory with surface effects: Application to columnar nanoporous materials

Nanocomposites, including nanoporous ceramics, metals and polymers, are being widely used in various engineering applications. At the nanoscale, surface/interface effects have significant influence on the homogenized properties and local stresses. To account for these nanoscale effects, the well-established Gurtin–Murdoch theory of surface elasticity is typically incorporated into classical elasticity-based micromechanics schemes and numerical approaches based on the finite-element and most recently finite-volume methods. Whereas the classical micromechanics models either neglect the interactions of adjacent fibers/pores or account for the interactions in an average sense, the finite-element and finite-volume based approaches require high discretization to capture the large stress/strain gradients around the interface/surface. In contrast, the recently proposed hybrid homogenization theory (HHT) combines elements of locally-exact elasticity and finite-volume approaches and is both accurate and efficient, enabling rapid generation and analysis of ordered and quasi-random fiber or pore arrays. Herein, a continuous bulk matrix layer of arbitrary size endowed with a two-dimensional energetic surface is introduced between fiber/pore and discretized matrix region that enables exact calculation of stress jump conditions in the Young–Laplace equations of the Gurtin–Murdoch model. The enhanced HHT with interface/surface effects is validated by comparison with the stress concentration factors obtained using a classical elasticity solution and point-wise stress fields predicted by the locally-exact and finite-volume homogenization theories, exhibiting stable solutions even for very thin bulk matrix layers. The enhanced theory is then employed to study the influence of random nanopore microstructures with random pore radii on the complete set of homogenized moduli of porous silicon used in the electronics industry.

On the propagation of plane waves in cubic quasicrystal plates with surface effects

In this paper, an exact solution of propagation characteristics of plane waves in an infinite cubic quasicrystal plate with surface effects has been derived. Based on the energy density of quasicrystal elastic deformation, the continuum theory of surface elasticity and new boundary conditions for quasicrystal surface layers are derived, from which we obtain an explicit closed form for dispersion relations of quasicrystal plates with the advanced boundary-value problems. The numerical results indicate that the overall stiffness of the quasicrystal plates strengthens with increasing surface elastic parameters and surface residual stresses, resulting in a continuous decrease in the phase velocity.

A novel constitutive model for surface elasticity at finite strains suitable across compressibility spectrum

The surface elasticity theory of Gurtin–Murdoch has proven to be remarkably successful in predicting the behavior of materials at the nano scale, which can be attributed to the fact that the surface-to-volume ratio increases as the problem dimension decreases. On the other hand, surface tension can deform soft elastic solids even at the macro scale resulting e.g. in elastocapillary instabilities in soft filaments reminiscent of Plateau–Rayleigh instabilities in fluids. Due to the increasing number of applications involving nanoscale structures and soft solids such as gels, the surface elasticity theory has experienced a prolific growth in the past two decades. Despite the large body of literature on the subject, the constitutive models of surface elasticity theory at large deformations are not suitable to capture the surface behavior from fully compressible to nearly incompressible elasticity, especially from a computational perspective. A physically meaningful and proper decomposition of the surface free energy density in terms of area-preserving and area-varying contributions remains yet to be established. We show that an immediate and intuitive generalization of the small-deformation surface constitutive models does not pass the simple extension test at large deformations and results in unphysical behavior at lower Poisson’s ratios. Thus, the first contribution of the manuscript is to introduce a novel decomposed surface free energy density that recovers surface elasticity across the compressibility spectrum. The second objective of this paper is to formulate an axisymmetric counterpart of the elastocapillary theory methodically derived from its three-dimensional format based on meaningful measures relevant to the proposed surface elasticity model. Various aspects of the problem are elucidated and discussed through numerical examples using the finite element method enhanced with surface elasticity.

Fracture analysis of circular hole edge arbitrary position crack with surface effects in magnetoelectroelastic materials

The mechanical properties of magnetoelectroelastic (MEE) materials are brittle, which is easy to produce nano-defects (hole or crack). The micro-fields of nanoscale cracks and holes are size dependent significantly. Nanoscale cracks and holes interact with each other, and the interference effect is very obvious. In this paper, the Model III fracture behavior of nanoscale circular hole edge arbitrary position radial crack is studied under anti-plane mechanical load, in-plane electric displacement load and in-plane magnetic induction load. Based on the surface elasticity theory, the conformal mapping technique and the MEE elasticity theory, the MEE fields expressions of the nano-hole and nano-crack are obtained, and the analytical solution of generalized intensity factors (stress intensity factor, electric displacement intensity factor and magnetic induction intensity factor) at crack tip is given. The comparison between the present results and the existing research shows the correctness of the present method in this paper. The influences of the microstructure and coupling loads on the generalized intensity factors are discussed.