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.     

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.     

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.     

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.     

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.     

承受双向等拉应力的平面应变I型裂纹线弹性断裂的I1准则与K准则一致性分析

采用线弹性有限元方法计算了承受双向等拉应力的平面应变I型裂纹的应力场,分析了裂纹尖端各应力分量间的关系,拟合了各非零应力分量关于裂纹半长度a和裂纹尖端最小网格尺寸l₁的函数,分析了应力第一不变量I₁与应力场强度因子K_I的相关性。结果表明,裂纹尖端各非零应力分量间存在稳定的比例关系;各非零应力分量值和加载应力的比值与裂纹半长度a的1/2次幂呈正比例关系、与裂纹尖端最小网格尺寸l₁的1/2次幂呈反比例关系;相同最小网格尺寸条件下,裂纹尖端的应力第一不变量与应力场强度因子的比值l₁/K_I为与加载应力和裂纹长度无关的常数,证明了承受双向等拉应力的平面应变I型裂纹线弹性断裂的I₁准则与K准则具有一致性。     

On the stress wave induced curving of fast running cracks — a numerical study by a time-domain boundary element method

Summary Fast crack propagation in dynamically loaded plane structures is investigated. The major point of interest is the evolution of the crack trajectory under the influence of stress waves which are generated and repeatedly reflected at the specimen boundaries. Since these waves may lead to arbitrary mixed-mode and time-dependent loading of the crack tip, both the direction and speed of crack advance are determined from a fracture criterion.Starting point is a system of time-domain boundary integral equations which describes the initial boundary value problem of a linear elastic body containing an arbitrarily growing crack. The unknown displacements and/or tractions on the exterior boundary and the displacement jumps across the crack are computed numerically by a collocation method in conjunction with a time-stepping scheme. Crack growth is modelled by adding new boundary elements of constant length at the running crack tip.The method proves to be of sufficient accuracy when applied to problems treated with other numerical techniques. Moreover, the simulation of dynamic crack propagation under various geometry and loading conditions enables the reproduction and analysis of complex phenomena observed experimentally.     

Analysis of a dynamically loaded three-point-bend ductile fracture specimen

The three-point-bend bar is a common specimen configuration used in experimental fracture studies. It is essentially a two-dimensional configuration in the form of a simply supported beam with an initial edge crack on the cross-section at mid-span. The specimen is loaded to fracture initiation by means of a concentrated transverse force applied at mid-span on the uncracked surface of the beam. In the present case, it is assumed that the material of interest is ductile, and that fracture initiation occurs after substantial plastic deformation, which develops in the uncracked ligament under the applied load. Furthermore, it is assumed that the loading is rapidly applied so that material inertia must be taken into account in relating applied loads to crack tip fields. It is supposed that the crack tip conditions are such that the J-integral may be adopted as a characterizing parameter. The main purpose of the present study is to determine conditions under which the value of J at initiation may be inferred from quantities that are directly measurable in an experiment. To this end, J is determined from computed field quantities by means of a crack tip integral that is suitable for finite element procedures. The value of J is simultaneously computed in terms of measurable quantities by means of an appropriate deep crack formula, and implications for fracture testing of tough materials at relatively high loading rates are discussed. The notion of a transition time, defined as the time beyond which a deep crack formula may be used to compute J in an experiment, is introduced on the basis of simple model studies. Calculations are performed for typical specimen dimensions and material properties representative of a high-strength structural steel in a ductile condition.     

Fracture analysis of dissimilar Al‐Al friction stir welded joints under tensile/shear loading

In this research, fracture of dissimilar friction stir welded (FSWed) joint made of Al 7075‐T6 and Al 6061‐T6 aluminum alloys is investigated in the cracked semi‐circular bend (CSCB) specimen under mixed mode I/II loading. Due to the elastic‐plastic behavior of the welded material and the existence of significant plastic deformations around the crack tip at the propagation instance, fracture prediction of the FSWed specimens needs some failure criteria in the context of the elastic‐plastic fracture mechanics which are very complicated and time‐consuming. For this purpose, the Equivalent Material Concept (EMC) is used herein by which the tensile behavior of the welded material is equated with that of a virtual brittle material. By combining EMC with the 2 brittle fracture criteria, namely the maximum tangential stress (MTS) and mean stress (MS) criteria, the load‐carrying capacity (LCC) of the FSWed CSCB specimens is predicted. Comparison of the experimental results and theoretical predictions from the 2 criteria showed that both criteria could accurately predict the LCC of the cracked specimens. Moreover, as the contribution of mode II loading increases, the size of the plastic region around the crack tip at failure increases, leading to increasing the LCC.     

Effect of material nonlinearity on crack propagation

A common test specimen for studying the dynamic characteristics of a crack is that of the so-called double cantilever beam. A starter crack is usually introduced at one end by a wedge action splitting the beam at the mid-section and producing an initial crack of certain length. Depending on the loading condition the crack first picks up speed and then slows down as it increases in length. This work is concerned with the effect of material nonlinearity on the motion of a running crack. An extensive amount of theoretical results are presented, with details of velocity and acceleration related to time and crack length parameters. The results are useful to the experimentalists for collecting data on the dynamic fracture of engineering materials.     

Numerical simulations of dynamic crack growth along an interface

Dynamic crack growth is analyzed numerically for a plane strain bimaterial block with an initial central crack. The material on each side of the bond line is characterized by an isotropic hyperelastic constitutive relation. A cohesive surface constitutive relation is also specified that relates the tractions and displacement jumps across the bond line and that allows for the creation of new free surface. The resistance to crack initiation and the crack speed history are predicted without invoking any ad hoc failure criterion. Full finite strain transient analyses are carried out, with two types of loading considered; tensile loading on one side of the specimen and crack face loading. The crack speed history and the evolution of the crack tip stress state are investigated for parameters characterizing a PMMA/Al bimaterial. Additionally, the separate effects of elastic modulus mismatch and elastic wave speed mismatch on interface crack growth are explored for various PMMA-artificial material combinations. The mode mixity of the near tip fields is found to increase with increasing crack speed and in some cases large scale contact occurs in the vicinity of the crack tip. Crack speeds that exceed the smaller of the two Rayleigh wave speeds are also found.     

Transient dislocation emission from a crack tip in an anisotropic elastic material

The emission of a dislocation with a general Burgers vector from the tip of a stationary semi-infinite crack in an anisotropic elastic material is examined. The dislocation is assumed to leave the crack tip along the crack extension plane at constant speed. Explicit expressions for the transient shielding stress intensity factors at the crack tip and the drag forces on the dislocations are derived. Numerical results for a class of cubic materials and two hexagonal crystals, zinc and cobalt, are given. Dislocation emission under plane stress wave loading is discussed.     

INTERFACIAL CRACK INITIATION UNDER QUASI-STATIC AND DYNAMIC LOADING CONDITIONS: AN EXPERIMENTAL STUDY

Abstract— Interfacial fracture parameters under quasi-static and dynamic loading are examined in a large elastic mismatch bimatenal system. A wide range of remote field loading ratios of shear and tension are considered. The crack tip fields are mapped using the optical method of coherent gradient sensing or CGS and fracture parameters are quantified. Distinctly different crack initiation responses are observed for positive and negative shear stresses acting on the interface. Also, low velocity impact loading experiments are conducted to study the influence of dynamic loading on crack initiation parameters. Dynamic interfacial crack tip fields are recorded using high speed photography and fracture parameters for dynamically loaded stationary cracks are obtained. Measurements suggest significant crack initiation toughness reduction under dynamic loading conditions.     

Influence of water on crack propagation in poly methyl methacrylate: craze stress and craze fibril lifetime

Polymethylmethacrylate (PMMA) is often used as a material in submarine applications. Therefore, the fracture properties of dry and wet PMMA in water and/or under hydrostatic pressure are of great importance. Previous work has shown that water strongly increases fracture toughness, and leads to a complicated figure of K ₁ versus crack speed, and stable-unstable crack and craze propagation, depending on external loading rate. In this study, compact tension specimens immersed in water have been tested on a tensile machine and crack tips have been observed during propagation by means of optical interferometry. Fracture stress intensity factors, and craze-zone shapes and sizes have been measured as a function of loading time and crack speed in water. The results have been rationalized in terms of craze fibril stress versus fibril extraction velocity and craze fibril lifetime versus fibril stress. Both may be expressed in terms of a stress-activated process governing fracture. It is found that, when expressed in these terms, the complicated influence of the external loading rate becomes irrelevant for describing local intrinsic material properties and K ₁ values. It is shown that there is no contradiction between the fact that water increases the fracture toughness, and the fact that the microscopic craze stress and craze fibril lifetime decrease at the crack tip.     

Dynamic fields generated by rapid crack growth

Stresses and strains near a rapidly propagating crack tip are affected by the mass density of the material. This paper starts with a brief summary of analytical results for near-tip dynamic fields as predicted by linear elastic fracture mechanics. Next, exact expressions are derived for dynamic crack-line strains, for mode-III crack propagation in a nonlinear elastic material and in an elastic perfectly-plastic material. These expressions are valid on the crack line from the moving crack tip to the moving boundary with the region of linearly elastic deformation. For steady-state crack growth, a critical strain criterion is used to compute the relation between external load and crack tip speed. The required external load increases with crack-tip speed.     

Numerical analysis of blunting of a crack tip in a ductile material under small-scale yielding and mixed mode loading

The blunting of the tip of a crack in a ductile material is analysed under the conditions of plane strain, small-scale yielding, and mixed mode loading of Modes I and II. The material is assumed to be an elastic-perfectly plastic solid with Poisson's ratio being 1/2. The stress and strain fields for a sharp crack under mixed mode loading are first determined by means of elastic-plastic finite element analysis. It is shown that only one elastic sector exists around the crack tip, in contrast with the possibility of existence of two elastic sectors as discussed by Gao. The results obtained for a sharp crack are used as the boundary conditions for the subsequent numerical analysis of crack tip blunting under mixed mode loading, based on slip line theory. The characteristic shapes of the blunted crack tip are obtained for a wide range of Mode I and Mode II combinations, and found to resemble the tip of Japanese sword. Also the stress field around the blunted crack tip is determined.     

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...     

Stability of dynamically propagating cracks in brittle materials

Summary Dynamic crack propagation and bifurcation phenomena are investigated analytically by utilizing the strain energy density fracture criterion in the framework of catastrophe theory. The effect of biaxial stress, loading imperfections (mixed-mode loading), Poisson's ratio, state of stress as well as crack tip propagation speed on the crack path directional stability is analyzed. Special crack path stability charts for (un)stably propagating cracks are obtained, and their connection with the experimentally recorded crack tip stress field is addressed. It is shown that a slight change of the normal stress acting parallel to a crack at its tip (crack-parallel stress) may be able to affect the crack surface roughening and/or branching velocity considerably. It is also indicated that under small tensile crack-parallel stress, the crack propagation is stable only when the crack propagation speed is less than about 30% of the relevant shear wave speed. The crack becomes unstable, and its surfaces roughen severely at a higher speed, and the crack bifurcates at the highest propagation speed, some 45% of the shear wave speed. It is suggested that superimposing mode-II (shear) loading will enhance the dynamic crack path stability while increasing crack propagation speed will reduce the stability of crack propagation. It is expected that under compressive crack-parallel stress no crack surface roughening will occur before the crack stably bifurcates.     

Mixed mode I/II fracture of 2024-T3 friction stir welds

For the first time, a series of mixed mode I/II fracture experiments have been performed on both base material and three families of friction stir welds (FSWs) in 6.4 mm thick, 2024-T351 aluminum plate; the FSW joints are designated hot, medium and cold due to the level of nominal weld energy input per unit weld length (specific weld energy) during the joining process.Results from the fracture tests indicate that the measured critical crack opening displacement (COD) at a fixed distance behind the crack tip properly correlates both load-crack extension response and microstructural fracture surface features for both the base metal and all FSWs, providing measure of a quantitative fracture toughness. The COD values also indicate that transition from mode I to mode II dominant crack growth occurs at lower loading angles for FSW joints having higher specific weld energy input, with a truly mixed mode I/II COD measured during crack growth in the medium FSW joint. Using results from recent detailed FSW metallographic studies, specific features in the fracture process are correlated to the FSW microstructure. Finally, the observed ductile crack growth path in all three welds tends to exit the under-matched FSW weld region as the far-field applied shear loading is increased, with the medium FSW being the only case where the flaw remained within the FSW region for all combinations of shear and tensile loading.     

Fracture Parameters for Natural Rubber Under Dynamic Loading

Abstract: An experimental study was conducted to evaluate the tear energy of unfilled and 25 phr carbon black‐filled natural rubber with varying loading rates. The variation of the tear energy with far‐field sample strain rate between 0.01 to 10 s^?1 was found to be different from tensile strip and pure shear specimens. Above a sample strain rate of 10 s^?1, the tear energy calculated from either specimen was comparable. The differences in the tear energy derived from the tensile strip and pure shear specimens were attributed to differences in the local crack tip stress state and strengthening of the material due to strain‐induced crystallisation. Both of these factors resulted in crack speeds 3–4 times higher in the pure shear specimen as compared to the tensile strip specimen. Finite element analysis (FEA) indicated that fracture would initiate at the crack tip either when the strain energy density approached the material toughness or when the maximum principal stress and strain approached the material tensile strength and fracture strain, respectively. It was concluded that these parameters would be better than the tear energy in predicting fracture of natural rubber under dynamic loading.