Effect of shear and rotary inertia on dynamic fracture of a beam or plate in tensile loading

The dynamic fracture response of a long beam of brittle material subjected to tensile loading is studied. If the magnitude of the applied tensile loading is increased to a critical value, a crack will propagate from one of the longitudinal surfaces of the beam. As an extension of previous work, the effect of shear and of rotary inertia on the tensile loading and the induced bending moment at the fracturing section is included in the analysis. Thus an improved formulation is presented by means of which the crack length, crack tip velocity, bending moment and axial force at the fracture section are determined as functions of time after crack initiation. It is found that the rotary effect diminishes the bending moment effect and retards total fracture time whereas the shear has an opposite effect. Thus by combining the two effects (to simulate to first order the Timoshenko beam) overall fracture is retarded. The results also apply for plane strain fracture of a plate in tensile loading provided the value of the elastic modulus is appropriately modified.     

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.     

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.     

Static analysis fatigue and fracture of cracked beams on elastic foundation

In this paper an analysis of a long edge-cracked beam resting on elastic foundation and with rotational and translational restraints at the ends is presented. The simplified Euler-Bernoulli beam theory is employed with the cracked section represented as an elastic hinge. By using the Paris-Erdogan crack propagation law, the fatigue life of the cracked beam is represented as a function of the constraints at the ends, the initial depth of the edge-crack, its location along the beam and the stiffness of the elastic foundation. The bending moment redistribution caused by the crack growth is analyzed and graphically illustrated. The retarding effect on the crack growth rate in the case of redundant structures subjected to repeated loading is discussed.     

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。      

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.     

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...     

Interface crack between two interface deformable piezoelectric layers

Based on an interface deformable piezoelectric bi-layer beam model, a bonded piezoelectric bi-material beam with an interface crack perpendicular to the poling axis is analyzed within the framework of the theory of linear piezoelectricity. The layer-wise approximations of both the elastic displacements and electric potential are employed, and each sub-layer is modeled as a single linearly elastic Timoshenko beam perfectly bonded together through a deformable interface. Using the impermeable crack assumption, the closed form solutions for the energy release rate (ERR) and crack energy density (CED) are derived for the layered piezoelectric beam subjected to combined uniformly distributed electromechanical loading. Based on superposition principle, both the ERR and CED and their components are all reduced to the functions of the crack tip loading parameters. Loading dependence of the total CED with respect to the applied electric field is manifested with the analytical results, showing that there is a transformation from an even dependence to an odd dependence for the normalized CED when the applied mechanical loading increases. Compared with the commonly used equivalent single layer model, the proposed analysis augments the crack driving force by alleviating the stress concentration along the interface and thus increases the loading parameters at the crack tip. The proposed model provides improved solutions for fracture analysis of piezoelectric layered structures and sheds light on the loading dependence of the fracture parameters (i.e., the ERR and CED) with respect to the applied electromechanical loadings.     

基于断裂力学的高强度钢材梁柱节点受力性能分析

为了研究国产Q460C高强度结构钢材梁柱节点的断裂行为,该文基于断裂力学理论,计算了Q460C高强度钢材焊缝及热影响区材料的断裂韧性,并且采用三维有限元断裂模型,以I型裂纹尖端应力强度因子K_I和J积分为定量的评价指标,分析了焊缝及热影响区不同长度的裂纹对梁柱节点断裂韧性的需求。弹性分析表明,K_I沿梁翼缘宽度方向呈W形分布,最大值出现在翼缘中心,且与名义弯曲应力呈线性关系,而焊根裂纹的断裂韧性需求比热影响区裂纹更高。弹塑性分析表明,J_I最大值出现在翼缘的边缘,热影响区裂纹的断裂韧性需求比焊根裂纹更高。研究结果表明,Q460C高强度钢材梁柱节点的断裂由焊根裂纹控制,断裂承载力与梁全截面塑性承载力相近,临界转角小于0.02rad,因此建议通过改善焊接工艺或局部构造来保证节点拥有足够的转动能力。     

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。     

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 transverse matrix cracking in cross-ply laminates loaded in simple bending

A one-dimensional analysis of a cross-ply laminate, containing cracked transverse plies, loaded in flexure is presented. Simple bending theory is used in conjunction with a shear-lag analysis, to calculate the degraded longitudinal modulus of a cracked transverse ply, enabling the flexural modulus of the laminate to be determined. The solution is shown to agree well with a more sophisticated stress transfer model in the literature. The analysis is then extended to calculate the applied bending moment at transverse crack onset under flexural loading using a fracture mechanics approach. The results suggest that the in situ transverse ply stress at which matrix cracking commences for the beam loaded in flexure is very close to the stress level at which the same ply would crack if the laminate were loaded in tension.     

Fracture behaviour of a TiAl alloy under various loading modes

Tensile tests, compression tests, in situ tensile tests, bending tests, tensile fatigue tests and bending fatigue tests were carried out for a TiAl alloy. Based on the global experimental results and microscopic observations of the fracture surfaces and cracking behaviour on the side surfaces of tested specimens, the fracture mechanisms of fully lamellar (FL) TiAl alloys under various loading modes are summarized as following: (1) Cracks initiate at grain boundaries and/or interfaces between lamellae. (2) When a crack extends to a critical length, which matches the fracture loading stress the crack propagates catastrophically through entire specimen. (3) The crack with the critical length can be produced promptly by the applied load in the tensile and bending test or be produced step-by-step by a much lower load in the fatigue tensile test. (4) For fatigue bending tests, the fatigue crack initiates and extends directly from the notch root, then extends step-by-step with increasing the fatigue bending loads. The fatigue crack maybe extends through entire specimen at a lower fatigue load or triggers the cleavage through the whole specimen at a higher load. (5) In compressive tests, cracks initiate and propagate in directions parallel or inclined to the compressive load after producing appreciable plastic strains. The specimen can be fractured by the propagation of cracks in both directions.     

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.     

Pre-cracking behaviour in the single cantilever beam adhesion test

The single cantilever beam adhesion test is a variant of the asymmetric wedge test in which a constant load is attached to the free adherend end to originate the moment required for joint fracture. It has the potential disadvantage of leading to an uncontrollable, accelerating crack, due to the constantly increasing applied couple, but presents the advantage of providing data on joint behaviour prior to crack initiation. It is this latter aspect that we consider in this paper. An apparently decreasing crack growth rate, as obtained from measurements of displacement of the free end of the beam is attributed to time-dependent adhesive strain. Use of classic, simple beam theory and Winkler, elastic foundation, equations allows us to assess an effectively static fracture energy, or fracture threshold.     

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.     

Fractography of unfilled and particulate-filled epoxy resins

The objective of this work was to analyse and understand the types of fracture surface morphology found in unfilled and particulate-filled epoxy resins in the light of the thermomechanical history of the specimen (loading rate or duration of loading, temperature, strain at break). Short-term tensile tests and long-term creep tests were conducted at four different temperatures. The fracture surface features were analysed using the scanning electron and optical microscopes and, where suitable, an image analyser. In order to correlate these morphologies with certain regimes of crack velocity, fracture mechanics tests were also conducted, varying the crack speed between 10^–7 and 10² m sec^–1. In the case of the filled resin, the lifetime under static loading is governed by a phase of slow, sub-critical crack growth which is manifested by resin-particle debonding. Thereafter, the crack accelerates and finally may reach terminal velocities depending on the amount of stored elastic energy available at the moment of fracture.     

An analysis of Chipping in Brittle Materials

The chipping process in a brittle material subjected to a uniformly applied edge load has been investigated. The present analysis extends earlier work by recognizing that as the chip is formed it may bend and change the loading at the crack tip. This geometry change introduces a nonlinear effect and has significant influence on the phenomenon. The nonlinear effect was demonstrated by incorporating it into an analytical model for a crack propagating along an interface parallel to the free surface. A finite-element analysis was then conducted to examine the crack trajectory formed in a homogeneous material. This numerical analysis showed that the crack reaches a maximum depth, and then deviates back to the free surface to form a spall. The form of this trajectory results from the additional bending moment acting at the crack tip induced by the bending of the chip and the consequent displacement of the applied load. The length of the spall was found to be approximately proportional to the square root of êd^5/2/K_IC, where K_IC is the fracture toughness of the material, ê is the appropriate modulus of the material, and d is the depth over which the edge load is applied. This revised version was published online in August 2006 with corrections to the Cover Date.     

Fracture initiation at a crack or notch in a viscoplastic material under high rate loading

The two-dimensional problem of an edge crack in a half space or plate is considered. The body is loaded by a suddenly applied, spatially uniform normal velocity imposed on the plane boundary of the body on one side of the edge crack. Otherwise, the boundary of the body, including the crack faces, is traction free. Both cases of an initially sharp crack tip and a narrow notch with small but nonzero notch root radius are considered. The material is modeled as elastic viscoplastic, including strain hardening, rate sensitivity and thermal softening. The applied loading produces predominantly mode II loading in the crack tip region. Under these conditions it is possible to nucleate an adiabatic shear band at the crack tip as a precursor to a mode II fracture. On the other hand, because of the rate sensitivity of the material and the high rate of loading, it may be possible under certain conditions to generate tensile stresses in the crack tip region sufficiently large to nucleate brittle tensile fracture. The problem is solved numerically by means of the finite element method in order to investigate the competition between these two possible fracture initiation mechanisms. The magnitude of the impact velocity imposed on the edge of the plate and the notch tip acuity have an effect on processes near the crack tip. For given material, the inception of crack growth is determined by the competition between a stress-based brittle fracture condition, associated with rate sensitivity and strain hardening, and a strain based criterion, associated with high strain rate and thermal softening.     

考虑应变梯度效应的三点弯梁模型解析研究--第一部分:局部化带的传播

为分析应变软化和由此带来的应变局部化问题,将梯度塑性理论引入裂纹带模型。以拉应变局部化区域代替裂纹带,在三点弯梁裂纹带(具有一定尺寸的带宽由特征长度确定)内部存在着不均匀分布的拉应变,这与实验结果相符。对拉应变进行积分,得到了拉应变局部化区域的张拉位移的理论表达式,结果表明:该位移与拉应力成线性规律,拉应变局部化区域的宽度越大,弹性模量越小或降模量越小,则该位移越大。此外,采用应力平衡条件、应变软化的本构关系及平截面假定,还得到了拉应变局部化区域的扩展规律,结果表明:下降模量越大、三点弯梁高度越小及弹性模量越小,则在相同的拉应力的情况下,拉应变局部化区扩展的长度越小;抗拉强度对拉应变局部化区扩展长度的最大值没有影响。此外,还研究了梁中部横截面内中性轴到具有最大承载能力的点的距离的变化规律。