Buckling and postbuckling of size-dependent cracked microbeams based on a modified couple stress theory

The elastic buckling analysis and the static postbuckling response of the Euler–Bernoulli microbeams containing an open edge crack are studied based on a modified couple stress theory. The cracked section is modeled by a massless elastic rotational spring. This model contains a material length scale parameter and can capture the size effect. The von Kármán nonlinearity is applied to display the postbuckling behavior. Analytical solutions of a critical buckling load and the postbuckling response are presented for simply supported cracked microbeams. This parametric study indicates the effects of the crack location, crack severity, and length scale parameter on the buckling and postbuckling behaviors of cracked microbeams.     

Stability and dynamic analysis of partially cracked thin orthotropic microplates under thermal environment: an analytical approach

Abstract

An analytical model is proposed to analyze the vibration and buckling problem of partially cracked thin orthotropic microplate in the presence of thermal environment. The differential governing equation for the cracked plate is derived using the classical plate theory in conjunction with the strain gradient theory of elasticity. The crack is modeled using appropriate crack compliance coefficients based on the simplified line spring model. The influence of thermal environment is incorporated in governing equation in form thermal moments and in-plane compressive forces. The governing equation for cracked plate has been solved analytically to get fundamental frequency and central deflection of plate. To demonstrate the accuracy of the present model, few comparison studies are carried out with the published literature. The stability and dynamic characteristics of the cracked plate are studied considering various parameters such as crack length, plate thickness, change in temperature, and internal length scale of microstructure. It has been concluded that the frequency and deflection are affected by crack length, temperature, and internal length scale of microstructure. Furthermore, to study the buckling behavior of cracked plate, the classical relations for critical buckling load and critical buckling temperature is also proposed considering the effect of crack length, temperature, and internal length scale of microstructure.     

DYNAMIC BUCKLING BEHAVIOR OF MULTI-WALLED CARBON NANOTUBES SUBJECTED TO STEP AXIAL LOADING

This paper studies the dynamic buckling behavior of multi-walled carbon nanotubes (MWNTs) subjected to step axial loading.A buckling condition is derived,and numerical results are presented for MWNTs u...     

Analysis of cracked functionally graded piezoelectric strip

The fracture behavior of a cracked strip under antiplane mechanical and inplane electrical loading is studied. A functionally graded piezoelectric strip with exponential material gradation is under consideration. The mechanical and electrical loading is combined via loading coupling factor. The problem of a graded piezoelectric strip containing a screw dislocation is solved. This solution results in stress and electric displacement components with Cauchy singularity. Based on the solution achieved for the dislocation, the distributed dislocation technique (DDT) is utilized to form any geometry of multiple cracks and analyze the behavior of a cracked strip under antiplane mechanical and inplane electrical loading. This technique is capable of the analysis of a strip with a system of interacting cracks. Several examples including strips with single crack, two straight cracks and two curved cracks are presented.     

Buckling of embedded multi-walled carbon nanotubes under combined torsion and axial loading

This paper describes an investigation into elastic buckling of an embedded multi-walled carbon nanotube under combined torsion and axial loading, which takes account of the radial constraint from the surrounding elastic medium and van der Waals force between two adjacent tube walls. Depending on the ratio of radius to thickness, the multi-walled carbon nanotubes discussed here are classified as thin, thick, and nearly solid. Critical buckling load with the corresponding mode is obtained for multi-walled carbon nanotubes under combined torsion and axial loading, with various values of the radius to thickness ratio and surrounded with different elastic media. The study indicates that the buckling mode (m, n) of an embedded multi-walled carbon nanotube under combined torsion and axial loading is unique and it is different from that with axial compression only. New features for the buckling of an embedded multi-walled carbon nanotube under combined torsion and axial loading and the meaningful numerical results are useful in the design of nanodrive device, nanotorsional oscillator and rotational actuators, where multi-walled carbon nanotubes act as basic elements.     

不均匀性焊接接头弹塑性断裂力学问题的有限元分析

本文对含裂纹焊接接头的简化模型——软夹硬不均匀裂纹体和硬夹软不均匀裂纹体——进行了弹塑性有限元分析,研究了材质力学性能不均匀性对OOD的影响规律.还研究了在静载和疲劳载荷下材质不均匀性对软夹硬不均匀体裂纹扩展行为的影响.     

Study on fracture behavior of ferroelectric ceramics under combined electromechanical loading by using a moiré interferometry technique

This paper discusses an in situ observation of fracture behavior around a crack tip in ferroelectric ceramics under combined electromechanical loading by use of a moiré interferometry technique. The deformation field induced by the electric field and the stress concentration near the crack tip in three-points bending experiments was measured. By analysis of the moiré images it is found that under a constant mechanical load, the electric field almost has no effect on the crack extension in the case that the directions of the poling, electric field and crack extension are perpendicular to each other. When the poling direction is parallel to the crack extension direction and perpendicular to the electric field, the strain decreases faster than that calculated by FEM with and without electrical loading as one goes away from the crack tip. In addition, as the electric field intensity increases, the strain near the crack tip increases, and the strain concentration becomes more significant. The project supported by the National Natural Science Foundation of China (10132010, 10025209, 10232023)     

Crack buckling in soft gels under compression

Recent interest in designing soft gels with high fracture toughness has called for simple and robust methods to test fracture behavior. The conventional method of applying tension to a gel sample suffers from a difficulty of sample gripping. In this paper, we study a possible fracture mechanism of soft gels under uni-axial compression. We show that the surfaces of a pre-existing crack, oriented parallel to the loading axis, can buckle at a critical compressive stress. This buckling instability can open the crack surfaces and create highly concentrated stress fields near the crack tip, which can lead to crack growth. We show that the onset of crack buckling can be deduced by a dimensional argument com- bined with an analysis to determine the critical compression needed to induce surface instabilities of an elastic half space. The critical compression for buckling was verified for a neo-Hookean material model using finite element simulations.     

开口复合材料柱壳屈曲与补救有限元分析

用有限元法对含有轴向裂纹的开口加强复合材料柱壳结构进行了补救研究;分析其压缩屈曲强度与模态情况,得到了裂纹长度与屈曲强度的关系,并与无裂纹的结构进行了对比.结果表明:裂纹长度在200mm以下时,对整个结构承载能力影响很小;当裂纹长度继续增大时,屈曲区域从开口上方转移到裂纹附近,屈曲强度开始急剧下降.为了加强裂纹所在区域结构的承剪能力,进行适当的修补后,可使屈曲模态与无裂纹柱壳相同,且屈曲强度稍有增加,从而证明了所提出补救方法的有效性.     

On the risk-based steam generator lifetime optimisation

A probabilistic fracture mechanics model is employed to estimate the failure probability of axially cracked steam generator tubes. The model estimates the failure probability from the random changes of the influencing parameters such as tube and crack geometry, material properties and non-destructive examination results, reliability and sizing accuracy and stable crack propagation. The performance of the model is illustrated by a numerical example. A steam generator tubing severely affected by the stress corrosion cracking is studied during most unfavourable accidental conditions. Two different plugging approaches are analyzed and the quality is compared, showing the superior performance of crack length oriented approach over tube wall thickness reduction both in terms of SG failure probability and extent of plugging. Thus, apart from setting the acceptable SG failure probability, all elements for the risk-based SG lifetime optimisation are provided on the example of stress corrosion cracking in the tube expansion transition zone.     

Wavy-edged fractures in axially split aluminium tubes

This paper is motivated by the somewhat unusual need to gain insight into the phenomenology of the mechanism by which a wavy edge is formed on the wreckage of some aircraft fuselage skins associated with aircraft destroyed in flight by on-board explosion.In order to explore the role of the plastic zone adjacent to the crack tip whilst avoiding the practical complications of generating the fractures explosively, simple quasi-static experiments have been carried out on aluminium tubes. Oversized rigid dies were pushed inside the tubes along their axes to generate fractures in Mode I and Mode III. It is conjectured that wavy edges are associated with fractures resulting from internal expansion of the tube by a travelling, internal, radial ring pressure region. The pressurised region behind the crack tip would be produced by explosively generated internal pressure being vented at the crack and, for the purpose of this study, is considered to be equivalent to that generated by the die. The production of such cracks is clearly demonstrated experimentally and contrasted with the plain-edge fractures produces during Mode III tearing fracture.A damage-model-based finite element analysis has been conducted to simulate the propagation of the crack and provide further insight into the strain and stress fields along the fractured edges. Both the experimental and numerical results show that this particular type of ring loading has to be applied to the tube to produce the wavy edge. Such a load expands the fractured flaps in the radial direction, stretching the material in the circumferential direction and, crucially, in the axial direction. The latter generates a relatively wide plastic wake close to and parallel to the fracture edge as the tube fails within which axial plastic strain predominates. Constrained by the remaining part of the tube that has not undergone plastic deformation, sufficient axial residual compressive stress can be produced in the plastic wake to produce a wavy edge which results from local buckling in the plastic wake. This mechanism suggests that ripples observed on the edges of fuselage skin wreckage are possible signatures of an internal explosion. The work described herein is also relevant to the deformation in a failed high-pressure gas pipe following the propagation of a ductile crack as noted previously in the literature.     

致密砂岩巷道模型试件动态起裂及止裂全过程分析

为了开展在不同冲击载荷作用下巷道围岩内裂纹的起裂、扩展及止裂等问题，以可调速冲击试验机进行动态加载试验，采用致密青砂岩制作裂纹巷道模型试件，并利用裂纹扩展计分别记录了动态起裂、扩展、止裂等时刻，对裂纹扩展速度的变化规律进行分析；随后采用AUTODYN有限差分法软件进行相应的数值模拟，数值模拟得到的裂纹扩展路径与试验结果基本一致。经过两者对比分析可知:随着冲击载荷作用的增加，裂纹平均扩展速度逐渐增大，随后趋于稳定值；预制裂纹的起裂时间随着冲击速度载荷的增加而逐渐降低，并在稳定值上下波动；随着冲击速度载荷的增加，裂纹扩展路径过程中的止裂时段逐渐变短。     

Plastic buckling of cylindrical shells under biaxial loading

The predictions for plastic buckling of shells are significantly affected by the plasticity model employed, in particular in the case of nonproportional loading. A series of experiments on plastic buckling of cylindrical aluminum alloy shells under biaxial loading (external pressure and axial tension), with well-defined loading and boundary conditions, was therefore carried out to provide experimental data for evaluation of the suitability of different, plasticity models. In the experiments, initial imperfections and their growth under load were measured and special attention was paid to buckling detection and load path control. The Southwell plot was applied with success to smooth the results. The results show that axial tension decreases resistance to buckling under external pressure in the plastic region due to softening of the material behavior. Comparison with numerical calculations usingJ ₂ deformation and incremental theories indicate that both theories do not predict correctly plastic buckling under nonproportional loading.Babcock (SEM Member), deceased, was Professor of Aeronautics and Applied Mechanics, California Institute of Technology, Pasadena, CA 91125.     

Length dependence of critical measures in single-walled carbon nanotubes

This paper presents an investigation on the buckling behaviour of single-walled carbon nanotubes under various loading conditions (compression, bending and torsion) and unveils several aspects concerning the dependence of critical measures (axial strain, bending curvature and twisting angle) on the nanotube length. The buckling results are obtained by means of an atomistic-scale generalized beam theory (GBT) that incorporates local deformation of the nanotube cross-section by means of independent and orthogonal deformation modes. Moreover, some estimates are also obtained by means of non-linear shell finite element analyses using Abaqus code. After classifying the buckling modes of thin-walled tubes (global, local and distortional), the paper addresses the importance of the two-wave distortional mode (flattening or ovalization mode) in their structural behaviour. Then, the well known expression to determine the critical strain of compressed nanotubes, which is based on Donnell theory for shallow shells, is shown to be inadequate for moderately long tubes due to warping displacements appearing in the distortional buckling modes. After that, an in-depth study on the buckling behaviour of nanotubes under compression, bending and torsion is presented. The variation of the critical kinematic measures (axial strain, bending curvature and twisting angle) with the tube length is thoroughly investigated. Concerning this dependence, some uncertainties that exist in the specific literature are meticulously explained, a few useful expressions to determine critical measures of nanotubes are proposed and the results are compared with available data collected from several published works (most of them, obtained from molecular dynamics simulations).     

Buckling stresses of composite tubes under biaxial compressive loads

An analytical model describing the instability of specially orthotropic composite tubes with geometric imperfections subject to biaxial compressive loads and under clamped-clamped boundary conditions is developed. Furthermore, the range of validity of the present solution is clarified, and comparisons are made to some studies on isotropic cylindrical shells. Six E-glass woven fabric-epoxy composite tubes with the same internal radius and different thicknesses and longitudinal lengths were fabricated and subjected to various combinations of external hydrostatic pressure and axial compressive load simultaneously. The normalized buckling stresses were found to agree in general with the theoretical predictions at various biaxial loadings. The buckling envelopes in normalization form provide useful design data on the strength of specially orthotropic composite tubes under a realistic range of biaxial loading conditions.     

冲击载荷下两裂纹间的连通规律

脆性材料内部含有大量裂纹，当某一裂纹扩展时，其他裂纹会对扩展裂纹产生影响。为了研究冲击载荷下，脆性材料内两裂纹的相互影响、连通规律及裂纹尖端应力强度因子的变化规律，利用有机玻璃板制作了含非平行双裂纹的实验试件，利用落板冲击设备进行了中低速冲击实验，结合有限元分析软件ABAQUS计算出裂纹尖端应力强度因子，利用有限差分软件AUTODYN进行了动态数值模拟研究，并将其模拟结果与实验结果进行对比分析。实验及模拟结果表明:裂纹破坏形态与AUTODYN数值模拟破坏形态基本一致；试件的断裂形态随着两裂纹间距不同而不同；裂纹间的相互影响程度随着裂纹间间距增大而减小；裂纹尖端应力强度因子K_I随着裂纹间距的增大而减小，而K_II随着裂纹间距增大而增大。     

The effect of load biaxiality on crack growth in non-linear materials

The influence of load biaxiality on the stress field and fracture behavior of a cracked plate is investigated. Considered is a square plate containing a central through the thickness crack and subjected to a biaxial loading perpendicular and parallel to the crack plane. The stress field of the plate is analyzed by a finite element code based on incremental plasticity and the von Mises yield condition. A method based on the strain energy density theory is used to determine the critical stress for crack initiation. It was found that the equi-biaxial loading mode induces the smallest plastic zones, while the critical applied stress for crack initiation becomes maximum. Quite the contrary happens for the shear loading system which causes the largest plastic zones and the minimum applied stress values fro crack growth. Results showing the dependence of the above quantities on the biaxiality of the applied stress are presented in graphical form.     

Prediction of the critical buckling load of multi-walled carbon nanotubes under axial compression

In this paper, we propose a new explicit analytical formula of the critical buckling load of double-walled carbon nanotubes (DWCNT) under axial compression. This formula takes into account van der Waals interactions between adjacent tubes and the effect of terms involving tube radii differences generally neglected in the derived expressions of the critical buckling load published in the literature. The elastic multiple Donnell shells continuum approach is employed for modelling the multi-walled carbon nanotubes. The validation of the proposed formula is made by comparison with a numerical solution. The influence of the neglected terms is also studied.     

一种随机裂纹结构的裂纹扩展路径分析方法

裂纹结构中存在大量不确定性因素,如裂纹长度、材料性质、外部载荷等,裂纹扩展路径的不确定性分析对研究随机裂纹结构损伤和断裂的力学特性并预测其性能及可靠性具有重要意义。本文提出了一种适应于混合载荷模式下随机裂纹结构的裂纹扩展路径分析方法。该方法考虑了裂纹长度、材料性质和外部载荷等的随机性,并通过蒙特卡洛方法对随机参数空间进行采样。采用比例边界有限元方法计算结构应力强度因子,进而模拟单次裂纹扩展路径。在此基础上,通过概率分析方法获得随机裂纹结构中裂纹扩展路径的统计特性。最后给出了两个数值算例验证了本文方法的有效性。     

The behavior of elastic tubes in a cohesion-free continuum

A model is made of a plain cross section of a tube (tunnel lining) in a cohesion-free continuum (rolling material, sand) which is represented by steel rollers of different diameters. The compactibility of sand in this model is represented by rubber inserts around the tunnel lining. The stress of the tube is measured by photoelasticity. These experiments are the start of a large program of investigation to calculate the stress deformation and buckling of elastic tubes under different loading conditions in rolling material.