Abschätzung von Bruchzähigkeitskennwerten aus der Bruch‐ oder Kerbschlagarbeit

Estimation of Fracture Toughness from the Charpy Fracture Energy By means of a simplistic mechanical model a mathematical relation between the total fracture energy of an edge‐cracked beam under bending and the fracture toughness was derived. Inserting the parameters of the standard Charpy specimens and accounting for the finite root radius and the shallowness of the notch, an equation to calculate fracture toughness from the Charpy fracture energy was obtained. Unlike the commonly used empirical correlation formulas, the presented equation is applicable to any elastic‐plastic material. From its theoretical basis and the underlying assumptions it is expected to be conservative in any case. Although the derivation only holds for the upper‐shelf regime, it also is applicable to the brittle‐to‐ductile transition regime as a lower bound. In this range, the degree of conservatism can be reduced by accounting for the well‐known shift of the transition temperature. Comparison with experimental data confirm these features of the presented formula.     

Ritz analysis of discontinuous beams using local trigonometric functions

The objective of the current paper is to present a Ritz-type analytical model for predicting the behavior of discontinuous beams such as thin-walled beams with cracks and multiply-stepped beams. The beam is discretized in the cracked as well as the un-cracked domains for a cracked thin-walled beam and in uniform beams for a multiple-stepped beam. A set of local trigonometric trial functions is used to define the twist angle for the cracked domain and the un-cracked domains, as well as to define the displacement field for uniform domains. A global equation system of unknown Ritz coefficients is derived by minimizing the Lagrangian functional or the total potential energy. In the present Ritz model, the interface continuity conditions between sub-domains are investigated and enforced into the global equation system using the condensation procedure or the Lagrange multipliers. Examples are presented to illustrate the effectiveness of the current model for free vibration and torsional analysis. Results obtained from the current model are found to agree well with those obtained using a detailed finite element method or with existing results in literature. The proposed model offers an efficient approach to reduce the modeling efforts and computational time required to analyze complex beams with cracks or multiple steps.     

Determination of the Stress Intensity Factor of an elliptical breathing crack in a rotating shaft

The failures due to the propagation of fatigue cracks are one of the most frequent problems in rotating machines. Those failures sometimes are catastrophic and are sufficient to provoke the loss of the complete machine with high risks for people and other equipments. When a cracked shaft rotates, the breathing mechanism appears. The crack passes from an open state to a close state with a transition in which a partial opening or closing of the crack is produced. In this work, a new general expression that gives the Stress Intensity Factor (SIF) along the crack front of an elliptical crack in a rotating shaft in terms of the crack depth ratio, the crack aspect ratio, the relative position on the front and the angle of rotation has been developed for linear elastic materials. By the moment, no expressions of the SIF in term of these variables have been found in the literature. To this end, a quasi-static 3D numerical model of a cracked shaft with straight and elliptical cracks subjected to rotary bending using the Finite Element Method (FEM) has been made. To simulate the rotation of the shaft, different angular positions have been considered. The SIF in mode I along the crack front has been calculated for each angular position of the cracked shaft and for different crack geometries. The expression results have been compared with solutions obtained from the literature. It has been found that they are in good agreement. The model has been applied to other crack geometries with good results. The obtained SIF expression allows studying the dynamic behavior of cracked shafts and can be used to analyze the crack propagation.     

An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using a nonlocal elasticity model

This paper is concerned with the free transverse vibration of cracked nanobeams modeled after Eringen's nonlocal elasticity theory and Timoshenko beam theory. The cracked beam is modeled as two segments connected by a rotational spring located at the cracked section. This model promotes discontinuities in rotational displacement due to bending which is proportional to bending moment transmitted by the cracked section. The governing equations of cracked nanobeams with two symmetric and asymmetric boundary conditions are derived; then these equations are solved analytically based on concerning basic standard trigonometric and hyperbolic functions. Besides, the frequency parameters and the vibration modes of cracked nanobeams for variant crack positions, crack ratio, and small scale effect parameters are calculated. The vibration solutions obtained provide a better representation of the vibration behavior of short, stubby, micro/nanobeams where the effects of small scale, transverse shear deformation and rotary inertia are significant.     

A mode II crack problem with sliding contact in an orthotropic strip

The problem of a centrally cracked, linear elastic orthotropic strip loaded in bending by three point forces is analyzed and discussed. Coulomb friction is assumed between the crack faces to study the influence of the friction coefficient on the strain energy release rate. Under certain simplifying assumptions the problem is reduced to the solution of a singular integral equation which is evaluated numerically. The results are compared with the solution of the same problem obtained using the beam theory; limits of the application of beam theory for the reduction of experimental data are discussed.     

Free transverse vibrations of cracked nanobeams with surface effects

The flexural vibrations of cracked micro- and nanobeams in the presence of surface effects are studied. The cracked-beam model is set up by dividing the classical cracked beam element into two segments connected by a rotational spring located at the cracked section. This model promotes a discontinuity in bending slope, which is proportional to the second derivative of the displacements. Numerical examples demonstrate the effects of beam length, and crack position and severity on the calculated values of natural frequencies of an anodic alumina nanowire in the presence of surface effects. Limiting cases are considered and good agreements with the data available in the literature are obtained.     

Novel beam analysis of end notched flexure specimen for mode-II fracture

A novel beam model of end notched flexure (ENF) specimen for mode-II fracture testing is presented. By applying the principle of superposition, the ENF specimen is modeled as two sub-problems: (1) an un-cracked beam under three-point bending; and (2) a skew symmetric cracked beam under shear traction on the crack surface. Due to skew-symmetry of sub-problem two, only the upper half of the beam is analysed, and based on compatibility of deformation, a shear compliance coefficient is introduced to establish beam deformation equation. Explicit and simple closed form solutions of compliance and strain energy release rate are obtained, and they compare well with existing finite element analyses. Compared to other available analytical methods of the ENF specimen, the present beam model is relatively simple and easy to use; further, it can be applied to other beam fracture specimen analysis (e.g., mixed mode fracture and bi-material interface specimen).     

On the analysis of a mixed mode bending sandwich specimen for debond fracture characterization

The mixed mode bending specimen originally developed for mixed mode delamination fracture characterization of unidirectional composites has been extended to the study of debond propagation in foam cored sandwich specimens. The compliance and strain energy release rate expressions for the mixed mode bending sandwich specimen are derived based on a superposition analysis of solutions for the double cantilever beam and cracked sandwich beam specimens by applying a proper kinematic relationship for the specimen deformation combined with the loading provided by the test rig. This analysis provides also expressions for the global mode mixities. An extensive parametric analysis to improve the understanding of the influence of loading conditions, specimen geometry and mechanical properties of the face and core materials has been performed using the derived expressions and finite element analysis. The mixed mode bending compliance and energy release rate predictions were in good agreement with finite element results. Furthermore, the numerical crack surface displacement extrapolation method implemented in finite element analysis was applied to determine the local mode mixity at the tip of the debond.     

New unloading compliance correlation for throughwall circumferentially cracked pipe to measure crack growth during fracture tests

Crack growth is generally measured during fracture experiment of specimen or component. The unloading compliance technique is commonly used for this purpose because of its simplicity. It infers the crack length from unloading compliance of cracked component. The pre‐requisite of this technique is the availability of an equation that correlates crack length with unloading compliance. While such correlations are available for compact tension and three‐point bend specimens, it is not available for big components such as pipe or pipe bend. Development of such a correlation for throughwall circumferentially cracked (TCC) straight pipe under bending, therefore, forms the objective of the present study. However, the challenge to develop such correlation for TCC pipe is that the equation should contain not only crack length as a function but also the current deformation or load level as a parameter. This is attributed to the fact that the circular cross section of the pipe ovalizes during deformation leading to change of bending stiffness of the cracked body. New compliance correlations have been proposed for TCC pipe under bending load considering these complexities. Elastic‐perfectly plastic material behaviour has been assumed to characterize the material stress–strain response. However, it has been shown that error due to this approximation with respect to the actual stress–strain behaviour is negligible if one chooses flow stress equal to average of yield and ultimate strength. The proposed correlations are expressed in terms of normalized parameters to make them independent of specific values of geometric dimensions such as radius, thickness and span length of four‐point bending loading system. Effectiveness of this normalization has also been verified by carrying out a sensitivity study.     

Dynamic stress intensity factors analysis of interface crack using line-spring model

In this study, a new method for calculating the dynamic stress intensity factors of a bimaterial bending specimen with an interface crack is proposed by making use of a line-spring model. A pre-cracked bending specimen is modeled by one-dimensional beam finite elements and a line-spring representing the stiffness or compliance of a cracked part. The proposed method enables the one-dimensional analysis of a two-dimensional crack problem; thus the time variations of the dynamic stress intensity factors of a bimaterial bending specimen with an interface crack can be obtained by making use of a personal computer within a few minutes. The results obtained by the proposed method agree reasonably well with those obtained by the two-dimensional finite element method, although a slight difference in period can be found. The proposed method enables rapid evaluation of dynamic stress intensity factors. So a rapid evaluation system of the dynamic fracture toughness of a bimaterial with an interface crack can be achieved by combining an instrumented impact test apparatus with a computer program based on the proposed method which runs on a personal computer.     

含裂纹平板的振动及裂纹扩展分析

针对含裂纹平板的振动与疲劳问题,研究含裂纹平板耦合动力学建模方法。首先在变形相似性原则下通过力学平衡原理推导出含裂纹项的平板振动方程,进而基于应力关系式形成裂纹项的表达式。在此基础上,利用Galerkin法将含裂纹板简化成单自由度振动系统,根据Berger经验方法产生方程的非线性项构建含裂纹板的非线性振动模型。最后通过算例探讨含裂纹板的动力学特性,协同Paris方程研究含裂纹平板动力学与裂纹扩展的耦合行为。研究结论表明,阻尼大小和激励力的变化对含裂纹板的振动特性及裂纹扩展规律具有显著影响。所提出的含裂纹平板耦合动力学建模方法,考虑了振动与裂纹扩展的耦合效应,为飞行器板结构抗振动疲劳设计提供理论依据。     

复合材料单面修补板裂纹尖端J积分的解析预测模型

考虑附加弯矩的影响,基于单搭接接头理论建立了单面修补含中心穿透裂纹直板解析模型,求解了修补结构基板的最大最小应力,并与有限元结果进行对比验证,研究了补片长度、宽度、厚度和胶层弹性模量对有限元模型裂纹尖端J积分的影响,通过拟合基板应力与有限元模型裂纹尖端J积分的数值关系,得到了求解修补结构裂纹尖端J积分的解析公式,并验证了其在单面修补弯曲板的适用性。通过研究和分析发现,求解的解析模型适用于承受面内载荷、面外载荷以及混合载荷下的平板和弯曲板修补结构。     

The creep of an elastoviscoplastic beam under a bending loading

Thermoplastics start to manifest a nonlinear mechanical behavior from relatively low loading levels. Under a bending solicitation, which generates a nonuniform stress field, the material behavior becomes more challenging. Indeed, a flexed specimen may have different behaviors from one point to another according to the local stress state. In the present work, a six-parameter rheological model is used to simulate the nonlinear behavior of an elastoviscoplastic beam, subjected to a three-point bending load. In the framework of Euler–Bernoulli theory, the mathematical formulation of a bent beam behavior involves the bending curvature function. This function allows the determination of the strain and stress fields along and through the beam. However, when the beam reaches the viscoplastic stage, the differential equation providing the bending curvature of the beam requires a numerical integration, which has been accomplished in this work. This theoretical modeling approach is supported by experimental creep tests carried out on polyamide specimens (PA6). The testing results are qualitatively consistent with the predictions of the proposed rheological model.     

Nonlinear non-classical microscale beams: Static bending, postbuckling and free vibration

This paper initiates the theoretical analysis of nonlinear microbeams and investigates the static bending, postbuckling and free vibration. The nonlinear model is conducted within the context of non-classical continuum mechanics, by introducing a material length scale parameter. The nonlinear equation of motion, in which the nonlinear term is associated with the mean axial extension of the beam, is derived by using a combination of the modified couple stress theory and Hamilton’s principle. Based on this newly developed model, calculations have been performed for microbeams simply supported between two immobile supports. The static deflections of a bending beam subjected to transverse force, the critical buckling loads and buckled configurations of an axially loaded beam, and the nonlinear frequencies of a beam with initial lateral displacement are discussed. It is shown that the size effect is significant when the ratio of characteristic thickness to internal material length scale parameter is approximately equal to one, but is diminishing with the increase of the ratio. Our results also indicate that the nonlinearity has a great effect on the static and dynamic behaviors of microscale beams. To attain accurate and reliable characterization of the static and dynamic properties of microscale beams, therefore, both the microstructure-dependent parameters and the nonlinearities have to be incorporated in the design of microscale beam devices and systems.     

弯曲梁柱结构平面外失稳的研究

研究了两节梁组成的弯曲梁在受竖向集中载荷作用下的平面外稳定问题,通过建立每节梁临界状态下弯曲和扭转变形微分方程,根据变形协调关系,得出了一端固定一端悬臂的弯曲梁平面外失稳的特征方程,并以等截面等长度梁构成的弯曲梁为对象,探讨了弯曲梁上拱和上翘对其平面外稳定性的影响,以及抗扭刚度对稳定性的影响。结果表明,同样的载荷作用幅度和梁高度,采用上拱的弯曲梁比直梁具有更高的侧向稳定性,并且存在一个最佳的弯曲角度,而上翘弯曲梁的侧向稳定性要低于直梁的侧向稳定性。当弯曲梁上翘时,增大抗扭刚度可以提高侧向稳定性,弯曲梁上拱时,增大抗扭刚度却会降低其侧向稳定性。     

Bending analysis of a finite beam on a reinforced halfplane

This paper deals with an experimental study, using the photoelastic method, on the bending analysis of a centrally loaded finite free-free beam resting on a partly reinforced (horizontally) elastic halfplane. In the experimental study the interface between the beam and partly reinforced halfplane has been made both frictionless and fully bonded. An experimental-numerical hybrid method, developed earlier by the authors has been used to determine the contact pressure distribution. The contact pressure distribution thus obtained has been compared with the results obtained using an analytical-numerical procedure in which two different modellings have been used for the reinforced region viz., (i) equivalent orthotropic (composite approach); and (ii) layered system (discrete approach). Results of contact pressure distribution as well as bending stress distribution in the beam have been presented for several cases. The experimental procedure presented here is easy to apply to such complex practical problems and could be of interest to design engineers.     

基于响应的梁损伤识别

采用扭转弹簧模拟悬臂梁的损伤，导出了损伤梁位移模态和转角模态的近似公式，获得了损伤梁在单点激励下，零初始条件的位移和转角响应。发现损伤梁的转角响应在损伤处发生阶跃变化，而损伤梁的转角响应对位置的一阶偏导数在损伤处有脉冲两数型突变的特点，从而提出了基于损伤梁转角响应的损伤判据函数。通过对损伤梁和损伤桁架结构的数值模拟，表明提出的判据函数不仅可以利用简谐激励下的响应识别梁的损伤，电可以利用冲击激励下的响应识别桁架的损伤。实验结果表明利用所提出的判据函数，可以同时判别损伤的位置和损伤的程度。     

含裂纹功能梯度Euler-Bernoulli梁和Timoshenko梁的屈曲载荷计算与分析

含裂纹构件的屈曲载荷是结构是否安全的判定准则之一, 其计算与分析也是结构健康监测和安全评价中关注的重要问题。基于Euler-Bernoulli梁理论和Timoshenko梁理论, 建立了一种求解含裂纹功能梯度材料梁的屈曲载荷计算方法。首先裂纹导致的构件截面转角不连续性由转动弹簧模型进行模拟, 再根据功能梯度材料Euler-Bernoulli梁和Timoshenko梁的屈曲控制方程及其闭合解, 由传递矩阵法建立了求解含裂纹功能梯度材料梁在多种边界条件下屈曲载荷的循环递推公式和特征行列式, 使问题通过降阶的方法得到快速准确的解答。数值算例研究了剪切变形、 裂纹的不同数目及位置、 材料参数变化、 长细比和不同边界约束条件等对含裂纹功能梯度材料梁屈曲载荷的影响。结果表明该方法可以简单、 方便和准确地计算不同数目裂纹和任意边界条件下功能梯度材料梁的屈曲问题。      

Dynamic response of a delaminated composite beam with general lay-ups based on the first-order shear deformation theory

The dynamic response analysis of a delaminated composite beam with a general lay-up traversed under an arbitrary moving/non-moving force is presented. By employing the energy method and introducing a new finite element, the global mass and stiffness matrices for a Laminated Composite Beam (LCB) of Timoshenko type are derived in which the material couplings (bending–tension, bending–twist, and tension–twist couplings) with the Poisson’s effect are considered. In deriving the governing equation the non-penetration condition is imposed by employing the method of Lagrange multipliers. Out of a self-developed finite element program, the natural frequencies and time response of such LCB are obtained. To check on the accuracy of the derived equation and hence, developed program, the obtained results are compared with the results from other available references out of which very good agreements are observed. Finally, by changing the laminate’s lay-ups, beam geometrical parameters and type of external force, the LCB’s natural frequencies and time responses are obtained as a primary base for the structural design.     

整体-局部弯曲模型及其在简支组合梁中的应用

为建立钢-混凝土组合梁界面滑移及考虑界面滑移的挠曲线计算模型,基于Bernoulli梁理论和剪力连接件线性剪力-滑移模型,从梁体平衡方程、几何方程和物理方程出发,建立了基于全微分的截面弯矩和剪力分配计算方法,提出了适用于组合梁变形分析的整体-局部弯曲模型,通过引入整体弯矩分配系数,建立了组合梁界面滑移及挠度控制方程。将整体-局部弯曲模型应用于简支组合梁变形分析,利用MAPLE软件对其进行求解,得到了均布荷载、跨中集中荷载、对称集中荷载三种荷载工况下的简支组合梁界面滑移公式与挠曲线方程,与国内外已有试验结果和相关研究成果相对比,结果表明该文计算结果与实际情况吻合较好。