It is showed that all equations of the linearized Gurtin-Murdoch model of surface elasticity can be derived, in a straightforward way, from a simple second-order expression for the ratio of deformed surface area to in...It is showed that all equations of the linearized Gurtin-Murdoch model of surface elasticity can be derived, in a straightforward way, from a simple second-order expression for the ratio of deformed surface area to initial surface area. This elementary derivation offers a simple explanation for all unique features of the model and its simplified/modified versions, and helps to clarify some misunderstandings of the model already occurring in the literature. Finally, it is demonstrated that, because the Gurtin-Murdoch model is based on a hybrid formulation combining linearized deformation of bulk material with 2nd-order finite deformation of the surface, caution is needed when the original form of this model is applied to bending deformation of thin-walled elastic structures with surface stress.展开更多
Analysis of the mechanical behavior of nanos- tructures has been very challenging. Surface energy and non- local elasticity of materials have been incorporated into the traditional continuum analysis to create modifie...Analysis of the mechanical behavior of nanos- tructures has been very challenging. Surface energy and non- local elasticity of materials have been incorporated into the traditional continuum analysis to create modified continuum mechanics models. This paper reviews recent advancements in the applications of such modified continuum models in nanostructures such as nanotubes, nanowires, nanobeams, graphenes, and nanoplates. A variety of models for these nanostructures under static and dynamic loadings are men- tioned and reviewed. Applications of surface energy and nonlocal elasticity in analysis of piezoelectric nanomateri- als are also mentioned. This paper provides a comprehensive introduction of the development of this area and inspires fur- ther applications of modified continuum models in modeling nanomaterials and nanostructures.展开更多
Within the context of Gurtin-Murdoch surface elasticity theory,closed-form analytical solutions are derived for an isotropic elastic half-plane subjected to a concentrated/uniform surface load.Both the effects of resi...Within the context of Gurtin-Murdoch surface elasticity theory,closed-form analytical solutions are derived for an isotropic elastic half-plane subjected to a concentrated/uniform surface load.Both the effects of residual surface stress and surface elasticity are included.Airy stress function method and Fourier integral transform technique are used.The solutions are provided in a compact manner that can easily reduce to special situations that take into account either one surface effect or none at all.Numerical results indicate that surface effects generally lower the stress levels and smooth the deformation profiles in the half-plane.Surface elasticity plays a dominant role in the in-plane elastic fields for a tangentially loaded half-plane,while the effect of residual surface stress is fundamentally crucial for the out-of-plane stress and displacement when the half-plane is normally loaded.In the remaining situations,combined effects of surface elasticity and residual surface stress should be considered.The results for a concentrated surface force serve essentially as fundamental solutions of the Flamant and the half-plane Cerruti problems with surface effects.The solutions presented in this work may be helpful for understanding the contact behaviors between solids at the nanoscale.展开更多
Considerations of nonlocal elasticity and surface effects in micro-and nanoscale beams are both important for the accurate prediction of natural frequency. In this study, the governing equation of a nonlocal Timoshenk...Considerations of nonlocal elasticity and surface effects in micro-and nanoscale beams are both important for the accurate prediction of natural frequency. In this study, the governing equation of a nonlocal Timoshenko beam with surface effects is established by taking into account three types of boundary conditions： hinged–hinged, clamped–clamped and clamped–hinged ends. For a hinged–hinged beam, an exact and explicit natural frequency equation is obtained. However, for clamped–clamped and clamped–hinged beams, the solutions of corresponding frequency equations must be determined numerically due to their transcendental nature. Hence, the Fredholm integral equation approach coupled with a curve fitting method is employed to derive the approximate fundamental frequency equations, which can predict the frequency values with high accuracy. In short,explicit frequency equations of the Timoshenko beam for three types of boundary conditions are proposed to exhibit directly the dependence of the natural frequency on the nonlocal elasticity, surface elasticity, residual surface stress, shear deformation and rotatory inertia, avoiding the complicated numerical computation.展开更多
The contact of a rigid body with nominally flat rough surface and an elastic half-space is considered.To solve the contact problem,the Greenwood-Williamson statistical model and the localization principle are used.The...The contact of a rigid body with nominally flat rough surface and an elastic half-space is considered.To solve the contact problem,the Greenwood-Williamson statistical model and the localization principle are used.The developed contact model allows us to investigate the surface approach and the real contact area with taking into account the asperities interaction.It is shown that the mutual influence of asperities changes not only contact characteristics at the macroscale,but also the contact pressure distribution at the microscale.As follows from the results,the inclusion in the contact model of the effect of the mutual influence of asperities is especially significant for studying the real contact area,as well as the contact characteristics at high applied loads.The results calculated according to the proposed approach are in a good agreement with the experimentally observed effects,i.e.,the real contact area saturation and the additional compliance exhaustion.展开更多
In order to ascertain the effects of atmospheric pressure on developmental characteristics and the stability of AEA(air-entraining agent)solution bubbles,AEA solution experiments and AEA solution bubble experiments we...In order to ascertain the effects of atmospheric pressure on developmental characteristics and the stability of AEA(air-entraining agent)solution bubbles,AEA solution experiments and AEA solution bubble experiments were,respectively,conducted in Peking(50 m,101.2 kPa)and Lhasa(3,650 m,63.1 kPa).Surface tensions and inflection-point concentrations were tested based on AEA solutions,whilst developmental characteristics,thicknesses and elastic coefficients of liquid films were tested based on air bubbles of AEA solutions.The study involved three types of AEAs,which were TM-O,226A,and 226S.The experimental results show that initial sizes of TM-O,226A,and 226S are,respectively,increased by 43.5%,17.5%,and 3.8%.With the decrease of ambient pressure,the drainage rate and the drainage index of AEA solution bubbles increase.Interference experiments show that the liquid film thicknesses of all tested AEA solution bubbles are in micron scales.When the atmospheric pressure decreases from 101.2 to 63.1 kPa,the liquid film thicknesses of three types of AEA solutions decrease in various degrees;and film elasticities at critical thicknesses increase.Liquid film of 226S solution bubbles is the most stable,presenting as a minimum thickness variation.It should be noted that elastic coefficient of liquid film only represents the level at critical thickness,thus it can not be applied as the only evaluating indicator of bubble stability.For a type of AEA,factors affecting the stability of its bubbles under low atmospheric pressure include initial bubbles size,liquid film thickness,liquid film elasticity,ambient temperature,etc.展开更多
A nonlocal elastic micro/nanobeam is theoretically modeled with the consideration of the surface elasticity, the residual surface stress, and the rotatory inertia,in which the nonlocal and surface effects are consider...A nonlocal elastic micro/nanobeam is theoretically modeled with the consideration of the surface elasticity, the residual surface stress, and the rotatory inertia,in which the nonlocal and surface effects are considered. Three types of boundary conditions, i.e., hinged-hinged, clamped-clamped, and clamped-hinged ends, are examined. For a hinged-hinged beam, an exact and explicit natural frequency equation is derived based on the established mathematical model. The Fredholm integral equation is adopted to deduce the approximate fundamental frequency equations for the clamped-clamped and clamped-hinged beams. In sum, the explicit frequency equations for the micro/nanobeam under three types of boundary conditions are proposed to reveal the dependence of the natural frequency on the effects of the nonlocal elasticity, the surface elasticity, the residual surface stress, and the rotatory inertia, providing a more convenient means in comparison with numerical computations.展开更多
The size-dependent elastic property of rectangular nanobeams （nanowires or nanoplates） induced by the surface elas- ticity effect is investigated by using a developed modified core-shell model. The effect of surface...The size-dependent elastic property of rectangular nanobeams （nanowires or nanoplates） induced by the surface elas- ticity effect is investigated by using a developed modified core-shell model. The effect of surface elasticity on the elastic modulus of nanobeams can be characterized by two surface related parameters, i.e., inhomogeneous degree constant and surface layer thickness. The analytical results show that the elastic modulus of the rectangular nanobeam exhibits a distinct size effect when its characteristic size reduces below 1 O0 nm. It is also found that the theoretical results calculated by a mod- ified core-shell model have more obvious advantages than those by other models （core-shell model and core-surface model） by comparing them with relevant experimental measurements and computational results, especially when the dimensions of nanostructures reduce to a few tens of nanometers.展开更多
Based on the variational principle, a continuum theory of surface elasticity and new boundary conditions for qua- sicrystals is proposed. The effect of the residual surface stress on a decagonal quasicrystal that is w...Based on the variational principle, a continuum theory of surface elasticity and new boundary conditions for qua- sicrystals is proposed. The effect of the residual surface stress on a decagonal quasicrystal that is weakened by a nanoscale elliptical hole is considered. The explicit expressions for the hoop stress along the edge of the hole are obtained using the Stroh formalism. The results show that the residual surface stress and the shape of the hole have a significant effect on the elastic state around the hole.展开更多
This study shows that it is possible to develop a well-posed size-dependent model by considering the effect of both nonlocality and surface energy,and the model can provide another effective way of nanomechanics for n...This study shows that it is possible to develop a well-posed size-dependent model by considering the effect of both nonlocality and surface energy,and the model can provide another effective way of nanomechanics for nanostructures.For a practical but simple problem (an Euler-Bernoulli beam model under bending),the ill-posed issue of the pure nonlocal integral elasticity can be overcome.Therefore,a well-posed governing equation can be developed for the Euler-Bernoulli beams when considering both the pure nonlocal integral elasticity and surface elasticity.Moreover,closed-form solutions are found for the deflections of clamped-clamped (C-C),simply-supported (S-S) and cantilever (C-F) nano-/micro-beams.The effective elastic moduli are obtained in terms of the closed-form solutions since the transfer of physical quantities in the transition region is an important problem for span-scale modeling methods.The nonlocal integral and surface elasticities are adopted to examine the size-dependence of the effective moduli and deflection of Ag beams.展开更多
A naturally discrete nanobar implies that the continuum axiom fails, and the surface-to-volume ratio is very large. The nonlocal theory of elasticity releasing the continuum axiom and the surface theory of elasticity ...A naturally discrete nanobar implies that the continuum axiom fails, and the surface-to-volume ratio is very large. The nonlocal theory of elasticity releasing the continuum axiom and the surface theory of elasticity are therefore employed to model tensile nanobars in this work. As commonly believed in the current practice, the axial nonlocal effect is only taken into account to analyze the mechanical behaviors of nanobars, regardless of the three-dimensional inherent atomistic interactions. In this study,a three-dimensional nonlocal constitutive law is developed to model the true nonlocal effect of nanobars, and based on which,a self-consistent variational bar model is proposed. It has been revealed for the first time how both the cross-sectional nonlocal interactions and the axial nonlocality affect the tensile behaviors of nanobars. It is found that the nonlocal influence predicted by the currently axial nonlocal bar model is grossly underestimated. Both the nonlocal cross-sectional and axial interactions become significant when the length-to-height ratio of nanobars is small. If the length-to-height ratio is relatively large(slender bars), the main nonlocal effect stems, however, from the nonlocal cross-sectional effect, rather than the axial nonlocal effect. This work also shows that it is possible to overcome the ill-posed problem of the pure nonlocal integral elasticity by employing both the pure nonlocal integral elasticity and surface elasticity. A well-posed size-dependent governing equation has been established for modeling nanobars under tension, and closed-form solutions are derived for their displacements. Based on the closed-form solutions, the effective elastic modulus is obtained and will be useful for calibrating the physical quantities in the "discretecontinuum" transition region for a span-scale modeling approach. It is shown that the effective elastic modulus may be softening or hardening, depending on the competition between the surface(modulus-hardening) and nonlocal(modulus-softening) effe展开更多
基金Financial support of the Natural Science and Engineering Research Council (NSERC)
文摘It is showed that all equations of the linearized Gurtin-Murdoch model of surface elasticity can be derived, in a straightforward way, from a simple second-order expression for the ratio of deformed surface area to initial surface area. This elementary derivation offers a simple explanation for all unique features of the model and its simplified/modified versions, and helps to clarify some misunderstandings of the model already occurring in the literature. Finally, it is demonstrated that, because the Gurtin-Murdoch model is based on a hybrid formulation combining linearized deformation of bulk material with 2nd-order finite deformation of the surface, caution is needed when the original form of this model is applied to bending deformation of thin-walled elastic structures with surface stress.
基金project was supported the National Natural Science Foundation of China (Grant 11372086)the Natural Science Foundation of Guangdong Province of China (Grant 2014A030313696)
文摘Analysis of the mechanical behavior of nanos- tructures has been very challenging. Surface energy and non- local elasticity of materials have been incorporated into the traditional continuum analysis to create modified continuum mechanics models. This paper reviews recent advancements in the applications of such modified continuum models in nanostructures such as nanotubes, nanowires, nanobeams, graphenes, and nanoplates. A variety of models for these nanostructures under static and dynamic loadings are men- tioned and reviewed. Applications of surface energy and nonlocal elasticity in analysis of piezoelectric nanomateri- als are also mentioned. This paper provides a comprehensive introduction of the development of this area and inspires fur- ther applications of modified continuum models in modeling nanomaterials and nanostructures.
基金supported by the National Natural Science Foundation of China(12272126,12272127)the Doctoral Fund of HPU(B2015-64).
文摘Within the context of Gurtin-Murdoch surface elasticity theory,closed-form analytical solutions are derived for an isotropic elastic half-plane subjected to a concentrated/uniform surface load.Both the effects of residual surface stress and surface elasticity are included.Airy stress function method and Fourier integral transform technique are used.The solutions are provided in a compact manner that can easily reduce to special situations that take into account either one surface effect or none at all.Numerical results indicate that surface effects generally lower the stress levels and smooth the deformation profiles in the half-plane.Surface elasticity plays a dominant role in the in-plane elastic fields for a tangentially loaded half-plane,while the effect of residual surface stress is fundamentally crucial for the out-of-plane stress and displacement when the half-plane is normally loaded.In the remaining situations,combined effects of surface elasticity and residual surface stress should be considered.The results for a concentrated surface force serve essentially as fundamental solutions of the Flamant and the half-plane Cerruti problems with surface effects.The solutions presented in this work may be helpful for understanding the contact behaviors between solids at the nanoscale.
基金the School of Civil and Environmental Engineering at Nanyang Technological University, Singapore for kindly supporting this research topic
文摘Considerations of nonlocal elasticity and surface effects in micro-and nanoscale beams are both important for the accurate prediction of natural frequency. In this study, the governing equation of a nonlocal Timoshenko beam with surface effects is established by taking into account three types of boundary conditions： hinged–hinged, clamped–clamped and clamped–hinged ends. For a hinged–hinged beam, an exact and explicit natural frequency equation is obtained. However, for clamped–clamped and clamped–hinged beams, the solutions of corresponding frequency equations must be determined numerically due to their transcendental nature. Hence, the Fredholm integral equation approach coupled with a curve fitting method is employed to derive the approximate fundamental frequency equations, which can predict the frequency values with high accuracy. In short,explicit frequency equations of the Timoshenko beam for three types of boundary conditions are proposed to exhibit directly the dependence of the natural frequency on the nonlocal elasticity, surface elasticity, residual surface stress, shear deformation and rotatory inertia, avoiding the complicated numerical computation.
基金supported the Russian Science Foundation(No.22-49-02010).
文摘The contact of a rigid body with nominally flat rough surface and an elastic half-space is considered.To solve the contact problem,the Greenwood-Williamson statistical model and the localization principle are used.The developed contact model allows us to investigate the surface approach and the real contact area with taking into account the asperities interaction.It is shown that the mutual influence of asperities changes not only contact characteristics at the macroscale,but also the contact pressure distribution at the microscale.As follows from the results,the inclusion in the contact model of the effect of the mutual influence of asperities is especially significant for studying the real contact area,as well as the contact characteristics at high applied loads.The results calculated according to the proposed approach are in a good agreement with the experimentally observed effects,i.e.,the real contact area saturation and the additional compliance exhaustion.
基金Funded by the National Natural Science Foundation of China(Nos.52178428,52178427,and 52308454)the Science and Technology Project of Tibet Department of Transportation(No.XZJTKJ[2020]04)。
文摘In order to ascertain the effects of atmospheric pressure on developmental characteristics and the stability of AEA(air-entraining agent)solution bubbles,AEA solution experiments and AEA solution bubble experiments were,respectively,conducted in Peking(50 m,101.2 kPa)and Lhasa(3,650 m,63.1 kPa).Surface tensions and inflection-point concentrations were tested based on AEA solutions,whilst developmental characteristics,thicknesses and elastic coefficients of liquid films were tested based on air bubbles of AEA solutions.The study involved three types of AEAs,which were TM-O,226A,and 226S.The experimental results show that initial sizes of TM-O,226A,and 226S are,respectively,increased by 43.5%,17.5%,and 3.8%.With the decrease of ambient pressure,the drainage rate and the drainage index of AEA solution bubbles increase.Interference experiments show that the liquid film thicknesses of all tested AEA solution bubbles are in micron scales.When the atmospheric pressure decreases from 101.2 to 63.1 kPa,the liquid film thicknesses of three types of AEA solutions decrease in various degrees;and film elasticities at critical thicknesses increase.Liquid film of 226S solution bubbles is the most stable,presenting as a minimum thickness variation.It should be noted that elastic coefficient of liquid film only represents the level at critical thickness,thus it can not be applied as the only evaluating indicator of bubble stability.For a type of AEA,factors affecting the stability of its bubbles under low atmospheric pressure include initial bubbles size,liquid film thickness,liquid film elasticity,ambient temperature,etc.
基金School of Civil and Environmental Engineering at Nanyang Technological University, Singapore for kindly supporting this research topic.
文摘A nonlocal elastic micro/nanobeam is theoretically modeled with the consideration of the surface elasticity, the residual surface stress, and the rotatory inertia,in which the nonlocal and surface effects are considered. Three types of boundary conditions, i.e., hinged-hinged, clamped-clamped, and clamped-hinged ends, are examined. For a hinged-hinged beam, an exact and explicit natural frequency equation is derived based on the established mathematical model. The Fredholm integral equation is adopted to deduce the approximate fundamental frequency equations for the clamped-clamped and clamped-hinged beams. In sum, the explicit frequency equations for the micro/nanobeam under three types of boundary conditions are proposed to reveal the dependence of the natural frequency on the effects of the nonlocal elasticity, the surface elasticity, the residual surface stress, and the rotatory inertia, providing a more convenient means in comparison with numerical computations.
基金Project supported by the National Natural Science Foundation of China (Grant No.11072104)the Scientific Research Program for Higher Schools of Inner Mongolia (Grant No.NJZY13013)
文摘The size-dependent elastic property of rectangular nanobeams （nanowires or nanoplates） induced by the surface elas- ticity effect is investigated by using a developed modified core-shell model. The effect of surface elasticity on the elastic modulus of nanobeams can be characterized by two surface related parameters, i.e., inhomogeneous degree constant and surface layer thickness. The analytical results show that the elastic modulus of the rectangular nanobeam exhibits a distinct size effect when its characteristic size reduces below 1 O0 nm. It is also found that the theoretical results calculated by a mod- ified core-shell model have more obvious advantages than those by other models （core-shell model and core-surface model） by comparing them with relevant experimental measurements and computational results, especially when the dimensions of nanostructures reduce to a few tens of nanometers.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11072104,1272053,and 11262017)the Key Project of the Chinese Ministry of Education(Grant No.212029)+3 种基金the Inner Mongolia Natural Science Foundation,China(Grant No.2013MS0114)the Natural Science Foundation of Inner Mongolia Department of Public Education,China(Grant No.NJZZ13037)Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region,China(Grant No.NJYT-13-B07)Program of Higher-Level Talents of Inner Mongolia University,China(Grant No.125125)
文摘Based on the variational principle, a continuum theory of surface elasticity and new boundary conditions for qua- sicrystals is proposed. The effect of the residual surface stress on a decagonal quasicrystal that is weakened by a nanoscale elliptical hole is considered. The explicit expressions for the hoop stress along the edge of the hole are obtained using the Stroh formalism. The results show that the residual surface stress and the shape of the hole have a significant effect on the elastic state around the hole.
基金Project supported by the National Natural Science Foundation of China(No.51605172)the Natural Science Foundation of Hubei Province of China(No.2016CFB191)the Fundamental Research Funds for the Central Universities(Nos.2722019JCG06 and 2015MS014)
文摘This study shows that it is possible to develop a well-posed size-dependent model by considering the effect of both nonlocality and surface energy,and the model can provide another effective way of nanomechanics for nanostructures.For a practical but simple problem (an Euler-Bernoulli beam model under bending),the ill-posed issue of the pure nonlocal integral elasticity can be overcome.Therefore,a well-posed governing equation can be developed for the Euler-Bernoulli beams when considering both the pure nonlocal integral elasticity and surface elasticity.Moreover,closed-form solutions are found for the deflections of clamped-clamped (C-C),simply-supported (S-S) and cantilever (C-F) nano-/micro-beams.The effective elastic moduli are obtained in terms of the closed-form solutions since the transfer of physical quantities in the transition region is an important problem for span-scale modeling methods.The nonlocal integral and surface elasticities are adopted to examine the size-dependence of the effective moduli and deflection of Ag beams.
基金support of the National Natural Science Foundation of China(Grant No.51605172)support of the Fundamental Research Funds for the Central Universities,Zhongnan University of Economics and Law(Grant No.2722020JCG060)the National Natural Science Foundation of China(Grant No.11801570)。
文摘A naturally discrete nanobar implies that the continuum axiom fails, and the surface-to-volume ratio is very large. The nonlocal theory of elasticity releasing the continuum axiom and the surface theory of elasticity are therefore employed to model tensile nanobars in this work. As commonly believed in the current practice, the axial nonlocal effect is only taken into account to analyze the mechanical behaviors of nanobars, regardless of the three-dimensional inherent atomistic interactions. In this study,a three-dimensional nonlocal constitutive law is developed to model the true nonlocal effect of nanobars, and based on which,a self-consistent variational bar model is proposed. It has been revealed for the first time how both the cross-sectional nonlocal interactions and the axial nonlocality affect the tensile behaviors of nanobars. It is found that the nonlocal influence predicted by the currently axial nonlocal bar model is grossly underestimated. Both the nonlocal cross-sectional and axial interactions become significant when the length-to-height ratio of nanobars is small. If the length-to-height ratio is relatively large(slender bars), the main nonlocal effect stems, however, from the nonlocal cross-sectional effect, rather than the axial nonlocal effect. This work also shows that it is possible to overcome the ill-posed problem of the pure nonlocal integral elasticity by employing both the pure nonlocal integral elasticity and surface elasticity. A well-posed size-dependent governing equation has been established for modeling nanobars under tension, and closed-form solutions are derived for their displacements. Based on the closed-form solutions, the effective elastic modulus is obtained and will be useful for calibrating the physical quantities in the "discretecontinuum" transition region for a span-scale modeling approach. It is shown that the effective elastic modulus may be softening or hardening, depending on the competition between the surface(modulus-hardening) and nonlocal(modulus-softening) effe
