Surface Elasticity Model Research Articles

On effective surface elastic moduli for microstructured strongly anisotropic coatings

The determination of surface elastic moduli is discussed in the context of a recently proposed strongly anisotropic surface elasticity model. The aim of the model was to describe deformations of solids with thin elastic coatings associated with so-called hyperbolic metasurfaces. These metasurfaces can exhibit a quite unusual behaviour and concurrently a very promising wave propagation behaviour. In the model of strongly anisotropic surface elasticity, strain energy as a function of the first and second deformation gradients has been introduced in addition to the constitutive relations in the bulk. In order to obtain values of surface elastic moduli, we compare dispersion relations for anti-plane surface waves obtained using the two-dimensional (2D) model and three-dimensional (3D) straightforward calculations for microstructured coatings of finite thickness. We show that with derived effective surface moduli, the 2D model can correctly describe the wave propagation.

Buckling of a stretched nanoplate with a nanohole incorporating surface energy

The effect of a nanoplate buckling at a circular nanohole under the remote uniaxial tension is studied incorporating the surface energy in accordance with the Gurtin–Murdoch surface elasticity model. The hole surface within the framework of the plane problem and the faces of the plate within the framework of the Kirchhoff theory of the plate bending are characterized by both the surface elasticity properties and the residual surface tension. The full potential energy of the plate containing the surface tension with non-strain terms of the surface displacement gradient is derived. Based on the principle of virtual displacements, critical values of the load, corresponding symmetrical and asymmetrical forms of the buckling are found by the Ritz method. Numerical investigations reveal that allowing for the non-strain terms of the surface displacement gradient in the Gurtin–Murdoch constitutive relation leads to the essential increasing the rigidity of the plate and the critical Euler load. Two types of the size effect are detected. The nanoplate thickness and the ratio of the hole radius to the thickness influence independently the value of the critical load.

Buckling of a nano‐rod with taken into account of surface effect

AbstractBuckling of a simple supported and cantilever nano‐rod and under action of self‐weight was modeled, taking into account the surface effect according to two models of surface elasticity – Gurtin–Murdoch and Steigmann–Ogden. The size effect under the buckling was studied by the Ritz method using two kinematic hypotheses, known as the rod theories of Euler–Bernoulli and Timoshenko. The simulation results were compared depending on the accepted surface elasticity model and the kinematic hypothesis.

Thermoelastic damping analysis of functionally graded sandwich microbeam resonators incorporating nonlocal and surface effects

Functionally graded (FG) sandwich microstructures/nanostructures stand as one of the most promising composite structures due to the capability to resist thermal shock and high noise in extreme thermal environments. Meanwhile, accurately prediction quality-factor based on thermoelastic damping (TED) is of great significance for the design of high-performance microresonators /nanoresonators. However, the classical TED models fail to explain the size-dependent thermo-mechanical behavior. The nonlocal elasticity and surface elasticity are responsible for the size-dependent phenomenon. In this article, a modified TED model is proposed to estimate the impact of the size-dependent effect on the TED of FG sandwich microbeam resonators in unsteady heat transfer processes by combining the nonlocal elasticity model, surface elasticity model, and dual-phase-lag (DPL) heat conduction model. It is assumed that the FG sandwich microbeam resonators consist of a ceramic core and FG surfaces. The energy equation and the transverse motion equation are derived. The analytical expression of TED is obtained by the complex frequency method. The influences of the factors including the nonlocal parameter, the surface effect, the power-law index, and the vibration modes on the TED are analyzed. These results provide a reasonable theoretical analysis to predicate TED in the design of FG sandwich microresonators/nanoresonators with high performance.

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.

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.

Multi-scale indentation model of stiff film-compliant substrate structures

As critical components or platforms, stiff film-compliant substrate structures are widely used in micro-electro-mechanical systems (MEMS) and other devices, in which their indentation behavior plays an important role. In this work, a multi-scale indentation model (MSI) for stiff film-compliant substrate structures is established based on the strain gradient theory and the surface elastic model. With this model, the indentation size effect of the structure is investigated from three aspects: the load-depth relationship, the distribution of bending moments in the film, and the surface morphology. The results show that the indentation size effect is mainly caused by the characteristic length and the surface residual stress of the film, and it is sensitive to the modulus ratio of the film to the substrate but has little relation with Poisson's ratio and the contact radius. Based on the above analysis, two dimensionless numbers are proposed to evaluate the indentation size effect of stiff film-compliant substrate structures. Besides, the predicted bending stiffness by the theoretical model is consistent with the experimental result, which verifies the effectiveness of the MSI model. Our research sheds light on the understanding of the indentation size effect of stiff film-compliant substrate structures and provides theoretical guidance for the design of related devices.

Circumferential Wrinkling of Elastic Cylinders With Negative Surface Tension

Abstract The present paper studies the critical condition for negative surface tension-driven circumferential wrinkling of soft cylinders based on the linearized Steigmann–Ogden model of surface elasticity. A simple negative surface tension-mode number relation is derived explicitly for arbitrary Poisson ratios of the cylinder and its surface layer and their shear modulus ratio, on which the critical surface residual strain and the associated mode number can be determined easily. For an incompressible solid cylinder with an incompressible thin surface layer, the critical values of surface residual strain and the mode number predicted by the present model are in good agreement with available numerical results based on the popular neo-Hooken nonlinear model for a wide range of material and geometrical parameters. In addition, the critical condition for circumferential wrinkling of the inner surface of a cylindrical hole within an infinite body is also derived. The present work addresses the key role of negative surface tension in circumferential wrinkling of soft cylinders and offers supporting evidence for the efficiency and accuracy of the linear Steigmann–Ogden model for the determination of the critical values for circumferential wrinkling.

Model III fracture analysis of a nanoscale elliptical hole with four cracks in one-dimensional hexagonal piezoelectric quasicrystals

When the defects are at the nanoscale, the failure characteristics of the piezoelectric quasicrystals are different from the macroscale case. The mode-III fracture problem of four nano-cracks emanating from a nanoscale elliptical hole in one-dimensional hexagonal piezoelectric quasicrystals was studied based on the Gurtin-Murdoch surface elasticity model. The analytical solutions of the electro-elastic field intensity factors and the energy release rate at the crack tip were obtained. The present solutions can be reduced to the existing results. The dimensionless electro-elastic field intensity factors show obvious size dependent effect. The influences of the defects size, the multi-cracks interaction, the phonon-phason field coupling coefficient and the applied electro-elastic loads on the fracture characteristics were discussed in detail.

Surface- and nonlocality-dependent vibrational behavior of graphene using atomistic-modal analysis

Classical continuum theories are incapable of accurately describing the behavior of nanostructures due to their size dependence. Size-dependent effects such as surface and non-local are among these phenomena, and their study at the nano-scale is essential. Despite advances in modeling and characterization of graphene's material properties, the surface and nonlocality effects of circular graphene nano-plates have not been calibrated for continuum mechanics theories. Based on a hybrid model that combines the advantages of continuum and atomic models, the present study is primarily focused on developing a method for calculating and estimating the effect of these parameters. A non-local surface elasticity model for graphene has been developed using a modal analysis derived from molecular dynamics simulations based on genetic algorithm optimization. In this research, both asymmetric and axisymmetric linear vibrational modes are studied. The parameters of the model, including surface residual stress, surface elasticity, surface density, and size parameter, are extracted from the results of the atomistic modal. Subsequently, this hybrid atomistic-continuum model has been employed to characterize the vibrational behavior of graphene. Based on the conducted simulations, the surface elasticity model with the nonlocality effect is calibrated with the lowest error for all radii and vibration modes. Moreover, this research suggests using a non-local surface elasticity model for very small radii since its findings are remarkably similar to the molecular dynamics simulations for the same diameters. On the other hand, the error is highest for the continuum model that does not incorporate surface and non-local effects. The findings of this study can be valuable in evaluating the crucial parameters regarding the surface and non-local elasticity of two-dimensional structures.

Size-dependent frequency of simply supported elastic ultra-thin films with surface effect under periodic vibration

Although surface effects play an important role in the mechanical properties of ultra-thin films, the nonlinear vibrations of ultra-thin films influenced by surface effects have not been fully understood. This paper develops an analytical framework for studying the nonlinear vibrations of simply supported ultra-thin films with surface effects. The framework is based on the modified Kirchhoff plate theory. The surface stress effects are treated by the Gurtin–Murdoch surface elasticity model and the motion equations include the effects of curvature and classical inertia. The dimensionless frequency of forcibly vibrated ultra-thin films with a simple support and surface effects is explicitly deduced through a series of perturbation procedure. Finally, the surface effects are evaluated in two numerical examples. In these demonstrations, the surface effects significantly influenced the dimensionless frequency when the film thickness reduced to one micrometer or less.

On the buckling and bending analysis of FG straight-sided quadrilateral nanoplates using a continuum mechanics-based surface elastic model

In this research, an elastic model based on the continuum mechanics is developed to study the static behaviors of functionally graded (FG) arbitrary straight-sided quadrilateral nanoplates. The model is constructed in the framework of Gurtin-Murdoch’s surface and Mindlin’s plate theories to account for the surface energy and shear deformation effects, simultaneously. The variational differential quadrature (VDQ) method is used along with a mapping technique to do the discretization process in a variational framework by means of differential and integral operators. Consequently, a weak form of governing equations is obtained from the energy quadratic representation of the problem. The solution method is of a distinguished feature as it involves just the first-order derivative of the field components in the mapping and discretization. After assuring the effectiveness of presented model by doing comparative studies, the critical buckling load and static deflection of the FG nanoplates with different shapes in geometry are investigated considering the surface effects. It is found that the surface energies effect on the static behavior of the rectangular nanoplates is more significant as compared to the non-rectangular nanoplates.

Surface effect on size dependent Young’s modulus of nanowires: Exponentially decreased surface elasticity model

Physical and mechanical properties of nanowires are strongly affected by surface effects. In this letter, a continuum theoretical scheme for describing the elastic property of nanowires was proposed. The influence of surface elasticity was considered as a decreasing regular pattern with increasing distance from surface. The decrease law was considered as exponential in this letter. An imaginary interface (between bulk like core and surface area) of nanowires appears in core-surface and core–shell models. This theoretical model avoided the non-physical interface between bulk like core and surface area of nanowires. The larger exponential decrease factor weakened surface effect obviously and was reflected in the effective Young’s modulus expression. Due to the influence of descending surface elasticity, the second, third, fourth and fifth surface modifications were introduced. This letter is aim to establish a continuum theory for elastic property of nanowires without any sudden change (interface) of Young’s modulus.

A New Approach for Testing Engagement Characteristics of Clutch

In this paper, a new method for testing the dynamic engagement characteristics of clutch is proposed. This approach is that the dynamic characteristics of clutch is simulated by finite element method. For the fluid friction characteristics, the average Reynolds equation proposed by Patir and Cheng is improved, and dimensionless parameters are adopted, which is applied to calculate the viscous torque. For the boundary friction characteristics, a surface elastic contact model is established to calculate rough contact torque. For the mixed friction characteristics, total torque consists of viscus torque and rough contact torque. The simulation and bench test are completed to verify the validity of model proposed. The model proposed can be used to guide the design of clutch in early stages of product development.

Wave propagation across a functionally graded interphase between soft and hard solids: Insight from a dynamic surface elasticity model

Joining soft to hard materials is a challenging problem in modern engineering applications. In order to alleviate stress concentrations at the interface between materials with such a mismatch in mechanical properties, the use of functionally graded interphases is becoming more widespread in the design of the new generation of engineered composite materials. However, current macroscale models that aim at mimicking the mechanical behavior of such complex systems generally fail in incorporating the impact of microstructural details across the interphase because of computational burden. In this paper we propose to replace the thin, but yet finite, functionally graded interphase by a zero-thickness interface. This is achieved by means of an original model developed in the framework of surface elasticity, which accounts for both the elastic and inertial behavior of the actual interphase. The performance of the proposed equivalent model is evaluated in the context of elastic wave propagation, by comparing the calculated reflection coefficient to that obtained using different baseline models. Numerical results show that our dynamic surface elasticity model provides an accurate approximation of the reference interphase model over a broad frequency range. We demonstrate application of this modeling approach for the characterization of the graded tissue system at the tendon-to-bone interphase, which fulfills the challenging task of integrating soft to hard tissues over a submillimeter-wide region.

Simulations versus tests for dynamic engagement characteristics of wet clutch

In this paper, the dynamic engagement characteristics of wet clutch are simulated by finite element method. In the fluid friction, the average Reynolds equation is amended and dimensionless parameters are involved, which is applied to calculate the viscous torque. In the boundary friction, a surface elastic contact model is established to calculate rough contact torque. In the mixed friction, total torque consists of viscus torque and rough contact torque. Experimental comparisons between the simulations and the SAE#2 bench tests are provide to verify the validity of the proposed method, the engagement time errors, the output torques maximum errors and the output torques average errors are utmost 4.86%, 3.87% and 0.73% respectively. The proposed method can be used to guide the design of wet clutches in early stages of product development.

A three-dimensional surface elastic model for vibration analysis of functionally graded arbitrary straight-sided quadrilateral nanoplates under thermal environment

AbstractIn this paper, a three-dimensional (3D) size-dependent formulation is developed for the free vibrations of functionally graded quadrilateral nanoplates subjected to thermal environment. The plate model is constructed within the frameworks of the Gurtin–Murdoch surface and the 3D elasticity theories. In this way, the effect of surface free energy and all the components of stress and strain tensors are considered without any initial assumption on them as there is no need to assume the variation of transverse normal stress inside the bulk material in advance. The variational differential quadrature approach and the mapping technique are applied to derive a discretized weak form of the governing equations. The present solution method bypasses the transformation and discretization of the higher order derivatives appearing in the equations of the strong form. The effects of surface stress, thermal environment, material gradient index and geometrical properties on the size-dependent vibrational behavior of quadrilateral nanoplates are investigated. It is observed that the thermal load intensifies the effect of surface free energy on the natural frequency of the nanoplates. The present model is exact in the extent of the continuum models and can be employed for structures with any thickness–span ratios.

On bending and buckling responses of perforated nanobeams including surface energy for different beams theories

Perforated beam is essential structural element of Nano-Electro-Mechanical-Systems (NEMS), whose design needs appropriate modelling of size of holes, hole numbers, and scale effects. The current manuscript presents a comprehensive study and develops non-classical closed form solutions to predict the static bending behavior and buckling stability of perforated nanobeams (PNBs) incorporating the surface energy for different four beams theories, for the first time. Equivalent geometrical models for both bulk and surface characteristics are presented. The Gurtin–Murdoch surface elasticity model is modified and adopted to include the perforation in surface characteristics. To consider the warping shear effect on bending as well as critical buckling loads with the presence of surface stress effects, four different beams theories with shear deformation are considered. The non-classical equilibrium equations relevant to each PNB theory are developed in detail. Closed-form solutions are developed considering the different classical and non-classical boundary conditions as well as loading conditions. The proposed methodology is verified by comparing the obtained results with the available analytical solutions for fully filled beams and excellent agreement is observed. Effects of perforation characteristics as well as the surface effects on bending and buckling behaviors are investigated.

Influence of frictional contact on indentation of elastic layer under surface energy effects

The paper examines the influence of frictional contact on an indentation problem of a finite-thickness elastic layer under a rigid flat-ended cylindrical indenter considering surface energy effects based on Coulomb friction model. The continuum-based approach corresponding to the complete version of Gurtin-Murdoch surface elasticity model is adopted to demonstrate the influence from surface energy. Normal and radial displacement boundary conditions in the contact region corresponding to the frictional indentation problem is formulated in the term of integral equations by employing the relevant displacement Green's function obtained by the use of Hankel integral transform. The contact tractions are solved by applying a collocation technique together with the assumption of a piecewise distribution for the tractions within each discretized annular ring element. The accuracy of proposed solution scheme is validated with existing solutions from classical continuum mechanics for frictional indentation problem of an elastic half-space. Selected numerical results are presented to demonstrate the influence of friction effects on the elastic fields of the layer under the influence of surface energy. It is found that the consideration of frictional contact makes the layer stiffer when compared to the frictionless indentation, and the influence of friction effect is smaller when the surface energy effects are accounted for. In addition, the elastic fields of the layer are clearly size-dependent and the elastic layer becomes stiffer with the presence of surface stresses.

Static bending of perforated nanobeams including surface energy and microstructure effects

This article aims to present comprehensive model and analytical solution to investigate the static bending behavior of regularly squared cutout perforated thin/thick nanobeams incorporating the coupled effect of the microstructure and surface energy for the first time. The perforation influence is considered to be deriving equivalent geometrical and material characteristics. The modified couple stress theory is adopted to incorporate the microstructure effect while the modified Gurtin–Murdoch surface elasticity model is employed to incorporate the surface stress effect in perforated nanobeams. A variational formulation based on minimization of the total potential energy principle is employed to derive the equilibrium equations of perforated nanobeams based on both Euler–Bernoulli and Timoshenko beams theories are developed to investigate the associated effect of the shear deformation due to perforation process. Additionally, Poisson’s effect is also incorporated. Analytical closed-form for the non-classical bending profiles as well as the rotational displacement are developed for both beam theories considering the simultaneous effect of both couple stress and surface stress for both uniformly distributed and concentrated loading patterns. The verification of the developed model is verified and compared with previous works, and an excellent agreement is obtained. The applicability of the developed model is demonstrated and applied to study and analyze the nonclassical bending behavior of regularly squared perforated simply supported beams under different loading conditions. Additionally, effects of the perforation configuration parameters, beam size as well as beam aspect ratio on the bending behavior of perforated beams in the presence of microstructure and surface stress effects are also investigated and analyzed. The obtained results reveal that both couple stress and surface stress significantly affect the bending behavior of regularly squared cutout perforated beam structures. Results obtained are supportive for the design, analysis and manufacturing of perforated NEMS applications.