Effect of bending moment on the dynamic fracture of a beam or plate under tensile loading

The dynamic fracture response of a long beam of brittle elastic material under tensile loading is studied. If the magnitude of the applied loading is increased to a critical value, a crack is assumed to propagate across the beam cross section. As an extension of previous work, an induced bending moment generated during fracture is incorporated into the analysis and this improved formulation is presented. The crack length, crack tip speed, axial force and bending moment on the fracturing section are determined as functions of time after crack initiation. It is found that the bending moment has a significant effect on the fracture process in that it tends to retard fracture and causes a drastic change in the slope of the loading curve for large crack depths. Finally, by appropriate change of the elastic modulus, the results may be applied to plane strain fracture of a plate in pure tensile loading.     

Dynamic fracture of a beam or plate under tensile loading

The dynamic fracture response of a long beam of brittle elastic material under tensile loading is studied. If the magnitude of the applied loading is increased to a critical value, a crack is assumed to propagate across the beam cross section. In a parallel analysis to [t] the crack length and applied loading at the fracture face are determined as functions of time measured from fracture initiation. The results of the analysis are shown in graphs of crack length, crack tip speed and fracturing section tensile loading vs time. As found in [1], the crack tip accelerates very quickly to a speed near the characteristic terminal speed for the material, travels at this speed through most of the beam thickness, and then decelerates rapidly in the final stage of the process. Finally, by appropriate change of the elastic modulus, the results may be applied to plane strain fracture of a plate under pure tensile loading.     

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.     

Stress analysis of the double cantilever beam specimen

A higher-order plate theory which includes transverse shear deformation and a thickness-stretch mode is utilized to analyze a complete double cantilever beam specimen. Homogeneous, orthotropic materials are considered. The beam is divided into a section along the crack and a second section along the uncracked region. Complete continuity of inplane force resultant, transverse shear force resultant, bending moment, and displacements are satisfied across the boundary between the two sections. This analysis allows one to obtain an approximate distribution of the interlaminar normal stress ahead of the crack. The effect of specimen geometry on energy release rate is investigated numerically. Consideration is also given to the average stress criterion as an alternative to a fracture mechanics approach for characterizing interlaminar peel strength.     

On the symmetric dynamic fracture of a beam under tensile loading

The dynamic fracture response of a long beam of brittle elastic material under tensile loading is studied by means of two different one-dimensional models. If the magnitude of the applied loading is increased quasi-statically to a critical value, two coplanar edge cracks are assumed to propagate across the beam's cross section. The first model parallels that of [6] with the crack length, crack speed and the loading on the fracturing section being determined as functions of time after fracture initiation. The second model is derived by means of energy considerations in the vicinity of the fracturing section. The results obtained from both models are similar except during the final phase of the fracture process.     

Dynamic response of clamped sandwich beam with aluminium alloy foam core subjected to impact loading

The dynamic response of clamped sandwich beam with aluminium alloy open-cell foam core subjected to impact loading is investigated in the paper. The face sheet and the core of the sandwich beam have the different thickness. And the sandwich beam is impacted by a steel projectile in the mid-span. The impact force is recorded by using accelerometer. The results show that tensile crack and core shear are the dominant failure modes. And the impact velocity and the thickness of the face sheet and the foam core have a significant influence on the failure modes and the impact forces. Combining with the inertia effect and experimental results, the failure mechanisms of the sandwich beams are discussed. The thickness of the foam core plays an important role in the failure mechanism of the sandwich beam. In present paper, the failure of the sandwich beam with a thin core is dominated by the bending moment, while the sandwich beam with a thick core fails by bending deformation in the front face sheet and the bottom face sheet in opposite direction due to the plastic hinges in the front face sheet.     

基于声发射技术的非标准自密实混凝土三点弯曲梁动态断裂特性

对不同初始缝高比的自密实混凝土(Self-compacting concrete,SCC)非标准三点弯曲梁开展不同加载速率下的断裂试验,获得其断裂的荷载-裂缝嘴张开口位移曲线及峰值荷载、断裂韧度、临界缝高比增量、弹性模量和柔度系数等断裂参数,结合Pearson相关性检验公式及加载速率效应模型,定量分析初始缝高比、加载速率与断裂参数间的相关性强弱及SCC断裂参数的加载速率效应。结果表明峰值荷载、断裂韧度及弹性模量均存在一定的加载速率效应,柔度系数仅与初始缝高比强相关,弹性模量和断裂韧度是材料的固有属性,不受初始缝高比影响。同时,基于声发射(Acoustic emission,AE)技术对SCC的损伤断裂过程、断裂边界效应及裂缝扩展模式进行分析,结果表明,AE参量能较好地反映混凝土断裂的三阶段特性及边界效应。裂缝的扩展首先以拉伸裂缝为主,剪切裂缝占比随着裂缝扩展过程逐渐增大。      

黏聚裂纹阻抗的弯曲梁承载力

在混凝土类软化材料断裂研究中,裂纹端部损伤区被简化为具有黏聚应力分布的非线性裂纹,该黏 聚力对裂纹扩展有阻抗作用。裂纹体的应力强度因子是断裂力学标志载荷作用与几何构型因素的量化表达指标; 黏聚力形成的阻抗强度因子数值,与黏聚裂纹长度和材料极值拉伸应力存在数量关系。通过双K断裂判据,以 带切口的三点弯曲梁为断裂力学模型,分析了裂纹黏聚阻力对断裂过程的影响规律,计算该弯曲梁结构断裂试 样的最大承担载荷;其结果比较符合实验数据。     

Shear fracture tests of concrete

Symmetrically notched beam specimens of concrete and mortar, loaded near the notches by concentrated forces that produce a concentrated shear force zone, are tested to failure. The cracks do not propagate from the notches in the direction normal to the maximum principal stress but in a direction in which shear stresses dominate. Thus, the failure is due essentially to shear fracture (Mode II). The crack propagation direction seems to be governed by maximum energy release rate. Tests of geometrically similar specimens yield maximum loads which agree with the recently established size effect law for blunt fracture, previously verified for tensile fracture (Mode I). This further implies that the energy required for crack growth increases with the crack extension from the notch. The R-curve that describes this increase is determined from the size effect. The size effect also yields the shear fracture energy, which is found to be about 25-times larger than that for Mode I and to agree with the value predicted by the crack band model. The fracture specimen is simple to use but not perfect for shear fracture because the deformation has a symmetric component with a non zero normal stress across the crack plane. Nevertheless, these disturbing effects appear to be unimportant. The results are of interest for certain types of structures subjected to blast, impact, earthquake, and concentrated loads.     

Effect of circular holes on the ratchet limit and crack tip plastic strain range in a centre cracked plate

In this paper a centre cracked plate subjected to cyclic tensile loading and cyclic bending moment is considered. The effect of circular holes drilled in the region of the crack tip on the ratchet limit and crack tip plastic strain range is studied. Direct evaluation of the ratchet limit and crack tip plastic strain range is solved by employing the new Linear Matching Method (LMM). Parametric studies involving hole diameter and location are investigated. The optimum hole location for reducing the crack tip plastic strain range with the least reduction in ratchet limit is identified, and located at a distance 10% of the semi-crack length from the crack tip on the side opposite the ligament for both cyclic tensile loading and cyclic bending moment cases. It is also observed that the optimum location is independent of the hole size for both cyclic loading cases.     

Shear deformation effect in second-order analysis of frames subjected to variable axial loading

In this paper a boundary element method is developed for the second-order analysis of frames consisting of beams of arbitrary simply or multiply connected constant cross section, taking into account shear deformation effect. Each beam is subjected to an arbitrarily concentrated or distributed variable axial loading, while the shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. To account for shear deformations, the concept of shear deformation coefficients is used. Three boundary value problems are formulated with respect to the beam deflection, the axial displacement and to a stress function and solved employing a BEM approach. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress function using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The influence of both the shear deformation effect and the variableness of the axial loading are remarkable.     

SHEAR FATIGUE CRACK GROWTH: A LITERATURE SURVEY

A fatigue crack is often initiated by a localized cyclic plastic deformation in a crystal where the active slip plane coincides with the plane of maximum shear stress. Once a crack is initiated, the crack will propagate on the maximum shear plane for a while and, in the majority of the cases, will eventually change to the plane of the applied tensile stress. The “shear” and “tensile” modes of fatigue crack propagation are termed stage I and stage II fatigue crack growth. They are also known as mode II and mode I fatigue crack growth. However, the mechanism of the tensile mode fatigue crack propagation is shear in nature. Considerable progress has been made recently in the understanding of mode II fatigue crack growth. This paper reviews the various test methods and related data analyses. The combined mode I and mode II elastic crack tip stress field is reviewed. The development and the design of the compact shear specimen are described and the results of fatigue crack growth tests using the compact shear specimens are reviewed. The fatigue crack growth tests and the results of inclined cracks in tensile panels, center cracks in plates under biaxial loading, cracked beam specimens with combined bending and shear loading, center cracked panels and the double edge cracked plates under cyclic shear loading are reviewed and analyzed in detail.     

A Displacement Solution to Transverse Shear Loading of Composite Beams by BEM

In this paper the boundary element method is employed to develop a displacement solution for the general transverse shear loading problem of composite beams of arbitrary constant cross section. The composite beam (thin or thick walled) consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli and are firmly bonded together. The analysis of the beam is accomplished with respect to a coordinate system that has its origin at the centroid of the cross section, while its axes are not necessarily the principal bending ones. The transverse shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. The evaluation of the transverse shear stresses at any interior point is accomplished by direct differentiation of a warping function. The shear deformation coefficients are obtained from the solution of two boundary value problems with respect to warping functions appropriately arising from the aforementioned one using only boundary integration, while the coordinates of the shear center are obtained from these functions using again only boundary integration. Three boundary value problems are formulated with respect to corresponding warping functions and solved employing a pure BEM approach. Numerical examples are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The accuracy of the obtained values of the resultant transverse shear stresses compared with those obtained from an exact solution is remarkable.     

The effects of strain rate and CCVC interfacial layer on mechanical behaviours of CALL

In this paper, the loading and loading-unloading tests of CALL and CALL (CCVC) under tensile impact have been carried out by a self-designed Rotating Circular Disk Tensile impact Apparatus. The quasi-static tension and short beam bending tests are performed on the Shimadzu-5000 testing apparatus. Experiment results show that both CALL and CALL (CCVC) have positive hybrid effect. Under quasi-static tension, the two composites have no obvious yielding until fracture, but have an obvious yielding point on the dynamic tensile stress-strain curves. The dynamic unstable fracture strain is about three times the static unstable fracture strain. The interlaminar shear strength (ISS) of CALL (CCVC) is 10 more than that of CALL. At the same time, the tensile strength and unstable fracture strain of CALL (CCVC) are also higher than that of CALL. In this paper, some conclusions are also drawn from the SEM observation of the fracture specimen surfaces.     

Confinement effect in flexural ductility of concrete: three-dimensional analysis

To study the effect of the transverse stress and strain distribution and steel ties (stirrups) upon the ultimate bending moment and bending ductility, a three-dimensional finite element analysis of a cross section slice is carried out. The slice consists of a layer of eight-node isoparametric elements, whose axial displacements are constrained so that the cross sections remain planar but not orthogonal. This allows interpreting the results in terms of curvature, bending moment, axial force and shear force. Each element within the layer is allowed to independently undergo cracking when its tensile strength limit is exceeded, and the incremental inelastic stiffness matrix of the cracked material is derived. The inelastic behavior of uncracked concrete or concrete between the cracks is modeled by the previously published endochronic theory, which allows representing the inelastic dilatancy due to shear, the hydrostatic pressure sensitivity, and the strain-softening (decrease of stress at increasing strain). The use of a constitutive relation that is capable of describing these effects is essential, since the dilatancy of concrete is opposed by ties which thus produce hydrostatic pressure in concrete thereby increasing its ductility. Transverse reinforcement is modeled either as reinforcement smeared throughout an element or as a steel bar connecting the nodes. Special measures are taken to eliminate spurious shear effects in the finite element model. A computer program to calculate the moment-curvature diagram of a given beam has been written using the incremental loading procedure. The calculated results compare satisfactorily with the available published test data on the effect of tie spacing upon the moment-curvature diagrams and flexural ductility.     

用能量方法研究混凝土断裂过程区的力学性能

准脆性混凝土自由裂缝前缘断裂过程区的发展与其非线性断裂特征及尺寸效应现象密切相关。它的物理力学行为的量化分析对理解混凝土断裂破坏机理和建立适用于混凝土结构裂缝稳定分析和安全评估断裂准则尤为重要,一直是混凝土断裂力学研究的核心问题。该文依据Hillerborg给出的断裂能定义,给出了计算单位长度断裂过程区发展能量耗散的通用表达式。以三点弯曲梁为例,采用非线性软化本构关系,进一步给出了计算此平均能量耗散的具体步骤及对应的公式。在根据实测的三点弯曲梁的断裂能回归拟合了特征裂缝张开位移w0后,计算了每个试件整个断裂全过程中不同荷载时刻断裂过程区耗能的平均值。结果表明:随着裂缝扩展,断裂过程区能量耗散的变化和试件尺寸无关,可描述断裂过程区混凝土材料的力学性能。     

应变率及CCVC界面层对CALL材料拉伸性能的影响

用自行研制的旋转盘式杆杆型冲击拉伸装置实施了ＣＦＲＰ／Ａｌ超混杂复合材料（ＣＡＬＬ）以及带有ＣＣＶＣ界面层的ＣＡＬＬ冲击拉伸加载及加卸载试验；用岛津试验机测定了它们的准静态拉伸及层间剪切性能材料在准静态拉伸时，应力－应变曲线无明显的屈服点，呈脆性断裂；而冲击拉伸时，应力－应变曲线有明显屈服点，呈脆－韧性断裂，动态失稳应变是准静态的３—４倍无论是准静态拉伸还是冲击拉伸，ＣＡＬＬ均存在明显的正混杂效应；ＣＡＬＬ（ＣＣＶＣ）层间剪切强度比ＣＡＬＬ约高１０％；在同一应变率下，ＣＡＬＬ（ＣＣＶ）的拉伸强度及失稳应变均比ＣＡＬＬ高．文中详细讨论以上试验现象，得出一些有意义的结论     

Plastic strains and the macroscopic critical plane orientations under combined bending and torsion with constant and variable amplitudes

Round cross-section specimens made of 18G2A steel were subjected to different combinations of constant- and variable-amplitude bending and torsion. The fatigue tests were performed under bending and torsion with moment control in the high cycle fatigue regime. Two approaches were used to calculate stress courses from moment histories. In one approach, stresses and strains were computed using simple elastic beam theory (nominal stresses). In the other approach, time courses of moments were used to calculate stress and strain histories taking into account plastic strains and non-linear stress distribution along the specimen cross-section on the basis of the algorithm described in the paper. The loading histories computed according the two methods were used to calculate the critical plane orientations. It was assumed that the orientation of the critical plane is controlled only by shear or tensile fatigue mechanism. Moreover, the theoretical critical plane positions were compared to the experimental macroscopic fatigue fracture plane orientations.     

Slice synthesis of a three dimensional “work of fracture” specimen

In the evaluation of brittle materials such as ceramics, the “work of fracture” specimen is enjoying considerable interest. That specimen, basically a beam bending specimen with rectangular cross section has inclined notches machined such that the remaining ligament is an isoceles triangle with the apex on the tension surface. The specimen is of particular interest because it tends to provide for slow stable crack propagation in such materials. The usual use of the test involves the recording of the load-deformation curve and the identification of the total area under the curve with the “work of fracture”. This then in turn is related to the fracture toughness.The present paper presents the results of an attempt to analyze this truly three dimensional problem in an approximate “two dimensional” fashion, treating the specimen as a series of slices and neglecting in effect the inter slice shear effects. Both beam bending and beam shear effects on compliance are considered. Plasticity effects are considered negligible for the class of brittle materials for which the test specimen is most frequently utilized. Comparison of experimental and analytical results is discussed. The observed stability characteristics of this “work of fracture” specimen is discussed in light of these results.     

陶瓷薄基板材料裂纹预制与断裂韧性评价

预制长度可控的裂纹以及原位观察裂纹扩展是研究陶瓷薄基板抗断裂行为的两大重点。本研究提出应变诱导法, 通过将基板与黄铜梁粘结形成复合体, 利用黄铜梁弯曲变形带动侧面陶瓷薄板受拉侧拉伸变形产生可控裂纹。在工具显微镜下对复合体进行四点弯曲, 原位观察样品的裂纹扩展情况, 通过调节黄铜梁宽度来控制初始裂纹长度, 在初始裂纹萌发后继续加载, 使裂纹达到测试断裂韧性的标准长度。将这种测试方法与块体材料断裂韧性的测试标准进行了对比, 结果表明: 采用该方法预制裂纹后测试断裂韧性具有简易性和可靠性。应变诱导法预制裂纹成功率高, 裂纹萌发位置及长度可控, 且操作方便, 可推广应用于超薄玻璃等其他超薄脆性材料的断裂韧性评价和分析裂纹扩展阻力。