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.

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 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.

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.

Vibration Analysis of a Cracked Beam on an Elastic Foundation

2016 ◽  
Vol 16 (05) ◽  
pp. 1550006 ◽  
Author(s):  
Ali Çağri Batihan ◽  
Fevzi Suat Kadioğlu
Keyword(s):  
Elastic Foundation ◽  
Cross Sections ◽  
Natural Frequencies ◽  
Crack Depth ◽  
Cracked Beam ◽  
Crack Location ◽  
Transverse Vibrations ◽  
Foundation Type ◽  
Foundation Parameters ◽  
Open Edge

The transverse vibrations of cracked beams with rectangular cross sections resting on Pasternak and generalized elastic foundations are considered. Both the Euler–Bernoulli (EB) and Timoshenko beam (TB) theories are used. The open edge crack is represented as a rotational spring whose compliance is obtained by the fracture mechanics. By applying the compatibility conditions between the beam segments at the crack location and the boundary conditions, the characteristic equations are derived, from which the nondimensional natural frequencies are solved as the roots. Sample numerical results showing the effects of crack depth, crack location, foundation type and foundation parameters on the natural frequencies of the beam are presented. It is observed that the existence of crack reduces the natural frequencies, whereas the elasticity of the foundation increases the stiffness of the system and thus the natural frequencies. It is also observed that the type of elastic foundation has a significant effect on the natural frequencies of the cracked beam.

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

2019 ◽  
Vol 19 (11) ◽  
pp. 1950127 ◽  
Author(s):  
Mohamed A. Attia ◽  
Rabab A. Shanab ◽  
Salwa A. Mohamed ◽  
Norhan A. Mohamed
Keyword(s):  
Surface Energy ◽  
Boundary Conditions ◽  
Nonlinear Vibration ◽  
Surface Stress ◽  
Couple Stress ◽  
Surface Elasticity ◽  
Functionally Graded ◽  
Residual Surface Stress ◽  
Stress Surface ◽  
Modified Couple Stress

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

Identification of Transverse Crack in Beam like Structure Using PSO

2014 ◽  
Vol 548-549 ◽  
pp. 1728-1734
Author(s):  
D.N. Thatoi ◽  
S. Choudhury ◽  
P.K. Jena ◽  
H.C. Das ◽  
A.K. Subudhi
Keyword(s):  
Natural Frequencies ◽  
Crack Depth ◽  
Population Based ◽  
Beam Structure ◽  
Cracked Beam ◽  
Swarm Optimization ◽  
Crack Location ◽  
Flexibility Matrix ◽  
Local Flexibility ◽  
Open Edge

The current proposed method has been developed using particle swarm optimization (PSO) technique. A single transverse open edge crack on a beam structure has been modeled using local flexibility matrix to determine natural frequencies. The PSO is a population based; bio-inspired evolutionary optimization algorithm that has been implemented for detection of crack. The frequencies obtained from analytical method have been used to train the PSO to get the desired output such as; relative crack depth and relative crack location. Mathematical modeling of the cracked beam structure is being done to ensure the integrity of the above algorithms. The results from the PSO show that both the size and location of the crack can be predicted efficiently through the proposed PSO.

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.

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.

scholarly journals Large Deformation and Instability of Soft Hollow Cylinder With Surface Effects

2021 ◽  
Vol 88 (4) ◽  
Author(s):  
Qitao Wang ◽  
Mingchao Liu ◽  
Zhenyu Wang ◽  
Changqing Chen ◽  
Jian Wu
Keyword(s):  
Surface Energy ◽  
Large Deformation ◽  
Hollow Cylinder ◽  
Surface Stress ◽  
Uniform Pressure ◽  
Pressure Loading ◽  
Residual Surface Stress ◽  
Buckling Behavior ◽  
Global Buckling ◽  
Hyper Elastic

Abstract Surface stress, which is always neglected in classical elastic theories, has recently emerged as a key role in the mechanics of highly deformable soft solids. In this paper, the effect of surface stress on the deformation and instability of soft hollow cylinder is analyzed. By incorporating surface energy density function into the constitutive model of a hyper-elastic theory, explicit solutions are obtained for the large deformation of soft hollow cylinder under the uniform pressure loading and geometric everting. The surface tension and the residual surface stress have a significant effect on the large deformation and instability of the soft cylinder. When the pressure loading and geometric everting are applied on the soft hollow cylinder, significant changes in the critical condition of the creases are found by varying the surface parameters. Two models of instability, surface crease and global buckling behavior, will be generated on the soft hollow cylinder with the uniform pressure, and the formed instability model is dependent on the ratio of the thickness to the radius. The results in this work reveal that surface energy obviously influences both the deformation and the instability of soft hollow cylinder at finite deformation and will be helpful for understanding and predicting the mechanical behavior of soft structures accurately.

Mismatch Effect on the Mixed Mode Type Creep Crack within Weldment

2016 ◽  
Vol 853 ◽  
pp. 281-285
Author(s):  
Jun Hui Zhang ◽  
Yan Wei Dai
Keyword(s):  
Mixed Mode ◽  
Stress Triaxiality ◽  
Crack Tip ◽  
Crack Tip Field ◽  
Creep Crack ◽  
Loading Mode ◽  
The Difference ◽  
Engineering Practices ◽  
Mismatch Effect ◽  
Transient And Steady State

Creep crack within weldments are very common in engineering practices, and the cracking location in these welding structures always appears at the HAZ location. The mismatch effect on the mixed mode creep crack is still not clear in these available literatures. The aim of this paper is to investigate the mismatch influence on the creep crack of mixed mode thoroughly. A mixed mode creep crack within HAZ is established in this paper. The leading factor that dominates the creep crack tip field under mixed loading mode is studied. The influences of mismatch effect on mode mixity, stress distribution and stress triaxiality are proposed. The difference of mixed mode creep crack and normal mode I or mode II creep crack are compared. The influence of mixity factor on the transient and steady state creep of crack tip are also analyzed.

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