Surface Effect and Size Dependence on the Energy Release Due to a Nanosized Hole Expansion in Plane Elastic Materials

2008 ◽  
Vol 75 (6) ◽  
Author(s):  
Q. Li ◽  
Y. H. Chen
Keyword(s):  
Energy Release ◽  
Surface Stress ◽  
Size Dependence ◽  
Surface Effect ◽  
High Surface ◽  
Plane Elasticity ◽  
Elastic Materials ◽  
Residual Surface Stress ◽  
Surface Lamé Constants ◽  
M Integral

This paper deals with the surface effect and size dependence on the M-integral representing the energy release due to a nanodefect expansion in plane elasticity. Due to the high surface-to-volume ratio for reinforcing particles in the nanometer scale, the surface effect along the nanosized hole may be induced from the residual surface stress and the surface Lamé constants. The invariant integrals such as the Jk-integral vector and the M-integral customarily used in macrofracture mechanics are extended to treat plane elastic materials containing a nanosized hole. It is concluded that both components of the Jk-integral vanish when the contour selected to calculate the integral encloses the whole nanosized hole. This leads to the independence of the M-integral from the global coordinate shift. It is concluded that the surface effect and the size dependence on the energy release due to the nanohole expansion are significant especially when the hole size is less than 40 nm. This present study reveals that the discrepancies of the M-integral value with the surface effect from the referenced value M0 without the surface effect are mainly induced from the residual surface stress τ0 rather than from the surface Lamé constants μs and λs.

Surface Effects on Large Deflection of Nanobeams Subjected to a Follower Load

2020 ◽  
Vol 12 (06) ◽  
pp. 2050067
Author(s):  
Yun Xing ◽  
Yi Han ◽  
Hua Liu ◽  
Jialing Yang
Keyword(s):  
Large Deformation ◽  
Surface Stress ◽  
Large Deflection ◽  
Surface Effect ◽  
Surface Elasticity ◽  
Surface Effects ◽  
Surface Residual Stress ◽  
Residual Surface Stress ◽  
Deformation Problem ◽  
Cross Sectional Dimension

As a basic element of the micro/nanodevices, nanobeams have remarkable physical properties and have attracted considerable attention in the previous studies. However, previous publications did not study the large deformation problem of nanobeams under follower loading when the surface effect becomes significant and especially for the influence of surface effect on mechanical behaviors of the nanobeams under follower loading remains unclear. In this paper, we investigated the large deformation behavior of nanobeams subjected to follower loads in consideration of the surface effects. The mechanical model of large deflection of extensible cantilever nanobeams under follower loading is presented in combination with the surface elasticity and residual surface stress, and then a MATLAB program of shooting method with a technique for determining the initial value was developed to solve the problems. The results indicate that the surface effects have an important influence on the large deflection of nanobeams under follower loading: when the surface residual stress is positive, the maximums of displacement in horizontal and vertical directions and the rotation angle of the free end become lager, but the corresponding follower force related to those maximums becomes smaller. When the residual surface stress is negative, the results are the opposite. In addition, the influence of the cross-sectional dimension of the nanobeams under follower loading on surface effects was discussed. This work is beneficial to understand the mechanism of large deformation of nanobeams with surface effects subjected to follower loads, and can also provide inspirations to design advanced nanomaterials and nanoscaled devices.

The M-Integral Analysis for a Nano-Inclusion in Plane Elastic Materials Under Uni-Axial or Bi-Axial Loadings

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
Tong Hui ◽  
Yi-Heng Chen
Keyword(s):  
Surface Effect ◽  
Continuum Theory ◽  
Elastic Materials ◽  
Hard Inclusion ◽  
Surface Parameters ◽  
Integral Analysis ◽  
Axial Tension ◽  
Soft Inclusion ◽  
Tension Loading ◽  
M Integral

This paper deals with the M-integral analysis for a nano-inclusion in plane elastic materials under uni-axial or bi-axial loadings. Based on previous works (Gurtin and Murdoch, 1975, “A Continuum Theory of Elastic Material Surfaces,” Arch. Ration. Mech. Anal., 57, pp. 291–323; Mogilevskaya, et al., 2008, “Multiple Interacting Circular Nano-Inhomogeneities With Surface/Interface Effects,” J. Mech. Phys. Solids, 56, pp. 2298–2327), the surface effect induced from the surface tension and the surface Lamé constants is taken into account, and an analytical solution is obtained. Four kinds of inclusions including soft inclusion, hard inclusion, void, and rigid inclusions are considered. The variable tendencies of the M-integral for each of four nano-inclusions against the loading or against the inclusion radius are plotted and discussed in detail. It is found that in nanoscale the surface parameters for the hard inclusion or rigid inclusion have a little or little influence on the M-integral, and the values of the M-integral are always negative as they would be in macroscale, whereas the surface parameters for the soft inclusion or void yield significant influence on the M-integral and the values of the M-integral could be either positive or negative depending on the loading levels and the surface parameters. Of great interest is that there is a neutral loading point for the soft inclusion or void, at which the M-integral transforms from a negative value to a positive value, and that the bi-axial loading yields similar variable tendencies of the M-integral as those under the uni-axial tension loading. Moreover, the bi-axial tension loading increases the neutral loading point, whereas the bi-axial tension-compression loading decreases it. Particularly, the magnitude of the negative M-integral representing the energy absorbing of the soft inclusion or void increases very sharply as the radius of the soft inclusion or void decreases from 5 nm to 1 nm.

Elastohydrodynamic Lubrication Line Contact Based on Surface Elasticity Theory

2020 ◽  
Vol 87 (8) ◽  
Author(s):  
Jie Su ◽  
Hong-Xia Song ◽  
Liao-Liang Ke
Keyword(s):  
Elastic Modulus ◽  
Film Thickness ◽  
Elasticity Theory ◽  
Surface Stress ◽  
Surface Effect ◽  
Surface Elasticity ◽  
Fluid Pressure ◽  
Elastohydrodynamic Lubrication ◽  
Line Contact ◽  
Residual Surface Stress

Abstract Using surface elasticity theory, this article first analyzes the surface effect on the elastohydrodynamic lubrication (EHL) line contact between an elastic half-plane and a rigid cylindrical punch. In this theory, the surface effect is characterized with two parameters: surface elastic modulus and residual surface stress. The density and viscosity of the lubricant, considered as Newtonian fluid, vary with the fluid pressure. A numerical iterative method is proposed to simultaneously deal with the flow rheology equation, Reynolds equation, load balance equation, and film thickness equation. Then, the fluid pressure and film thickness are numerically determined at the lubricant contact region. Influences of surface elastic modulus, residual surface stress, punch radius, resultant normal load, and entraining velocity on the lubricant film thickness and fluid pressure are discussed. It is found that the surface effect has remarkable influences on the micro-/nano-scale EHL contact of elastic materials.

Elastic Energy of Surfaces and Residually Stressed Solids: An Energy Approach for the Mechanics of Nanostructures

2015 ◽  
Vol 82 (1) ◽  
Author(s):  
Xiang Gao ◽  
Daining Fang
Keyword(s):  
Surface Energy ◽  
Elastic Energy ◽  
Surface Stress ◽  
Strain Energy ◽  
Surface Effect ◽  
Surface Elasticity ◽  
Small Deformation ◽  
Energy Approach ◽  
Residual Surface Stress ◽  
The Impact

The surface energy plays a significant role in solids and structures at the small scales, and an explicit expression for surface energy is prerequisite for studying the nanostructures via energy methods. In this study, a general formula for surface energy at finite deformation is constructed, which has simple forms and clearly physical meanings. Next, the strain energy formulas both for isotropic and anisotropic surfaces under small deformation are derived. It is demonstrated that the surface elastic energy is also dependent on the nonlinear Green strain due to the impact of residual surface stress. Then, the strain energy formula for residually stressed elastic solids is given. These results are instrumental to the energy approach for nanomechanics. Finally, the proposed results are applied to investigate the elastic stability and natural frequency of nanowires. A deep analysis of these two examples reveals two length scales characterizing the significance of surface energy. One is the critical length of nanostructures for self-buckling; the other reflects the competition between residual surface stress and surface elasticity, indicating that the surface effect does not always strengthen the stiffness of nanostructures. These results are conducive to shed light on the importance of the residual surface stress and the initial stress in the bulk solids.

Surface Effect on Static Bending of Functionally Graded Porous Nanobeams Based on Reddy’s Beam Theory

2019 ◽  
Vol 19 (06) ◽  
pp. 1950062 ◽  
Author(s):  
Jie Su ◽  
Yang Xiang ◽  
Liao-Liang Ke ◽  
Yue-Sheng Wang
Keyword(s):  
Surface Stress ◽  
Surface Effect ◽  
Beam Theory ◽  
Functionally Graded ◽  
Transverse Load ◽  
Transverse Displacement ◽  
Porosity Distribution ◽  
Residual Surface Stress ◽  
Porosity Coefficient ◽  
Distribution Type

In this paper, the surface effect on the static bending behavior of functionally graded porous (FGP) nanobeams subjected to a concentrated transverse load is studied by using Reddy’s higher-order beam theory. Three types of porosity distributions are considered for the nanobeam, i.e. uniform porosity distribution, symmetric and asymmetric non-uniform porosity distributions. With the consideration of the surface effect, the nanobeams can be abstracted as a composite beam composed of a surface layer and a bulk volume. According to the generalized Young–Laplace equation, the normal stress discontinuity across a surface due to the effect of surface stress is taken into consideration. The analytical solutions of the static bending problem of FGP nanobeams are obtained for the beams with hinged-hinged, clamped-clamped and clamped-free boundary conditions. The effects of the residual surface stress, porosity distribution type, porosity coefficient and length-to-thickness ratio on the transverse displacement of the FGP beams are discussed.

Transverse Vibrations of Mixed-Mode Cracked Nanobeams With Surface Effect

2015 ◽  
Vol 138 (1) ◽  
Author(s):  
Kai-Ming Hu ◽  
Wen-Ming Zhang ◽  
Zhi-Ke Peng ◽  
Guang Meng
Keyword(s):  
Surface Energy ◽  
Surface Stress ◽  
Mixed Mode ◽  
Natural Frequencies ◽  
Surface Effect ◽  
Crack Depth ◽  
Crack Tip ◽  
Residual Surface Stress ◽  
Transverse Vibrations ◽  
Local Flexibility

Slant edge cracked effect considering the inherent relation between surface energy and mixed-mode crack propagations on the free transverse vibrations of nanobeams with surface effect is investigated. First, the slant edge cracked effect, which considers residual surface stress effect on the crack tip fields of a mode-I and mode-II surface edge crack, is developed and the corresponding stress intensity factors (SIFs) and local flexibility coefficients are derived. Moreover, a refined continuum model of slant cracked nanobeams is established by considering both slant edge cracked effect and surface effect. The effects of fracture angles, crack depth, surface elasticity, surface stress, and surface density on the local flexibility and free transverse vibration characteristics of cracked nanobeams are, respectively, analyzed. The results show that the flexibility coefficients distribute symmetrically about residual surface stress. Fracture angles have a profound influence on both the symmetries of the mode shapes and the natural frequencies of nanobeams, and the influence becomes more pronounced as crack depth ratios increase. Furthermore, the natural frequencies will first decrease and then increase with fracture angles when the slant edge cracked effect is considered. The results demonstrate that the inherent relation between surface energy and crack propagations should be considered for both the stress distributions at the crack tip and the dynamic behavior of cracked nanobeams.

Two State M-Integral Analysis for a Nano-Inclusion in Plane Elastic Materials Under Uni-Axial or Bi-Axial Loadings

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
Tong Hui ◽  
Yi-Heng Chen
Keyword(s):  
Surface Tension ◽  
Dominant Role ◽  
Elastic Materials ◽  
Integral Analysis ◽  
Surface Lamé Constants ◽  
M Integral

In this paper, the two state M-integral is extended from macrofacture to nanodefect mechanics. The question as to why the M-integral for a nanovoid or a soft nano-inclusion might be negative is clarified. It is concluded that the surface tension plays a dominant role in evaluating the M-integral, whereas the surface Lamè constants yield much less influence than the surface tension. Their influence on the M-integral for a nanovoid or a soft nano-inclusion could be neglected.

scholarly journals Computation of Energy Release Rates for a Nearly Circular Crack

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
Nik Mohd Asri Nik Long ◽  
Lee Feng Koo ◽  
Zainidin K. Eshkuvatov
Keyword(s):  
Integral Equation ◽  
Energy Release Rate ◽  
Energy Release ◽  
Release Rate ◽  
Linear Equations ◽  
Circular Crack ◽  
Plane Elasticity ◽  
Hypersingular Integral Equation ◽  
Hypersingular Integral ◽  
Energy Release Rates

This paper deals with a nearly circular crack, in the plane elasticity. The problem of finding the resulting shear stress can be formulated as a hypersingular integral equation over a considered domain, and it is then transformed into a similar equation over a circular region, , using conformal mapping. Appropriate collocation points are chosen on the region to reduce the hypersingular integral equation into a system of linear equations with unknown coefficients, which will later be used in the determination of energy release rate. Numerical results for energy release rate are compared with the existing asymptotic solution and are displayed graphically.

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